MATHEMATICS | MOST PREDICTED QUESTIONS

CHAPTER 7 Tangents

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A ladder leans against a wall such that the top of the ladder is 12 units above the ground and the base of the ladder is 5 units from the wall. What is the length of the ladder?

Using the Pythagorean theorem, we get (ladder length)^2 = (height)^2 + (base distance)^2. So, ladder length = √((12)^2 + (5)^2) = √(144 + 25) = √169 = 13 units.

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If the distance between the centers of two circles is equal to the sum of their radii, the circles are ____.

If the distance between the centers of two circles is equal to the sum of their radii, the circles are touching externally. This means they touch at exactly one point.

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The common tangent to two circles that does not intersect the line segment joining their centers is called ____.

The common tangent to two circles that does not intersect the line segment joining their centers is called an external tangent. This type of tangent lies outside both circles and touches them at exactly one point each.

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A circle has a radius of 7 units. Calculate the length of a tangent from a point 25 units away from the center.

Using the Pythagorean theorem, we get (tangent length)^2 + (7)^2 = (25)^2. So, tangent length = √((25)^2 – (7)^2) = √(625 – 49) = √576 = 24 units.

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A line that touches a circle at exactly one point is called a ____.

A tangent is a line that touches a circle at exactly one point. This point is called the point of tangency. Unlike a secant which intersects the circle at two points, a tangent only touches the circle at one point. This unique property makes it a crucial concept in circle geometry. The tangent is perpendicular to the radius at the point of contact.

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In a cyclic quadrilateral, the sum of opposite angles is ____.

In a cyclic quadrilateral, the sum of the measures of opposite angles is always 180°. This is known as the supplementary property of cyclic quadrilaterals and is used in various geometric proofs and problems.

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If the distance between the centers of two circles is equal to the difference of their radii, the circles are ____.

If the distance between the centers of two circles is equal to the difference of their radii, the circles are touching internally. This means they touch at exactly one point inside the smaller circle.

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A line touches a circle at exactly one point. This line is called a ____.

A line that touches a circle at exactly one point is called a tangent. This point is called the point of tangency. The tangent is perpendicular to the radius at the point of contact.

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If the radius of a circle is 5 units, what is the length of a tangent from a point 13 units away from the center?

Using the Pythagorean theorem, we have (tangent length)^2 + (radius)^2 = (distance from center)^2. So, tangent length = √((13)^2 – (5)^2) = √(169 – 25) = √144 = 12 units.

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The angle subtended by a chord at the center of a circle is called ____.

The angle subtended by a chord at the center of a circle is called the central angle. This angle is significant in circle geometry because it is directly related to the arc it intercepts and the properties of the chord.

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If a wheel of a bicycle touches the ground at a point and the distance from the center of the wheel to the point of contact is 70 cm, what is this distance called?

The distance from the center of the wheel to the point of contact with the ground is called the radius. The radius of a circle is a line segment from the center to any point on the circumference, including the point where it touches the ground.

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In a right triangle, the circle that is tangent to all three sides is called ____.

The incircle of a triangle is a circle that is tangent to all three sides of the triangle. In a right triangle, the incircle touches each side at exactly one point. The radius of the incircle is related to the area and the semiperimeter of the triangle.

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A flagpole is 10 units tall. If a person standing 24 units away from the base of the flagpole observes the top of the pole, what is the length of the hypotenuse formed?

Using the Pythagorean theorem, we get (hypotenuse)^2 = (height)^2 + (distance from base)^2. So, hypotenuse = √((10)^2 + (24)^2) = √(100 + 576) = √676 = 26 units.

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In a circle, the tangent at any point is perpendicular to ____.

The tangent at any point on a circle is perpendicular to the radius at that point. This perpendicularity is a defining characteristic of tangents and is used to solve many geometric problems. It is also a key concept in understanding the properties of circles and their tangents.

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The length of the tangent from a point 10 units away from the center of a circle is 6 units. What is the radius of the circle?

Using the Pythagorean theorem, we get (tangent length)^2 + (radius)^2 = (distance from center)^2. So, radius = √((10)^2 – (6)^2) = √(100 – 36) = √64 = 8 units.

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A watchtower is 20 units high. If a person stands 21 units away from the base of the watchtower, what is the length of the hypotenuse formed between the top of the watchtower and the person?

Using the Pythagorean theorem, we get (hypotenuse)^2 = (height)^2 + (distance from base)^2. So, hypotenuse = √((20)^2 + (21)^2) = √(400 + 441) = √841 = 29 units.

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If two tangents are drawn to a circle from an external point, then ____.

When two tangents are drawn from an external point to a circle, they are equal in length. This property is a consequence of the fact that the tangents form two congruent right triangles with the radii to the points of tangency. The equality of these tangents is used in various geometric problems and proofs.

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A tree 9 units tall casts a shadow of 12 units. What is the length of the hypotenuse formed?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, hypotenuse = √((9)^2 + (12)^2) = √(81 + 144) = √225 = 15 units. This hypotenuse is the direct distance from the tree top to the shadow tip.

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If two tangents are drawn to a circle from an external point P, and the points of tangency are A and B, then the angle APB is ____.

When two tangents are drawn from an external point to a circle, the angle between the tangents at the external point (angle APB) is 90 degrees. This is because the tangents are equal in length and form two congruent right triangles with the radii to the points of tangency.

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A circle has a radius of 9 units. Calculate the length of a tangent from a point 15 units away from the center.

Using the Pythagorean theorem, we get (tangent length)^2 + (radius)^2 = (distance from center)^2. So, tangent length = √((15)^2 – (9)^2) = √(225 – 81) = √144 = 12 units.

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A statue 10 units tall casts a shadow of 7 units. What is the distance from the top of the statue to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((10)^2 + (7)^2) = √(100 + 49) = √149 = 12 units. This hypotenuse represents the direct distance from the statue top to the shadow tip.

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If the lengths of two tangents drawn from an external point to a circle are 9 units and 12 units, what can be concluded about the point?

If the lengths of two tangents drawn from an external point to a circle are given, the point lies outside the circle. This is because tangents can only be drawn from points outside the circle, and they are equal in length. However, there seems to be a misunderstanding in the question since both tangents should have equal lengths. The correct length should be equal.

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The angle between a tangent and a radius at the point of tangency is always ____.

The angle between a tangent and a radius at the point of tangency is always 90°. This perpendicularity is a fundamental property of tangents and circles.

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In a cyclic quadrilateral, the sum of the measures of the opposite angles is ____.

In a cyclic quadrilateral, the sum of the measures of the opposite angles is always 180°. This property is used in various geometric proofs and problems involving cyclic quadrilaterals.

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The point where a tangent touches a circle is called the ____.

The point where a tangent touches a circle is called the point of tangency. This point is significant in geometry because it is the only point where the tangent and the circle intersect, and it is where the radius is perpendicular to the tangent.

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The perpendicular drawn from the center of a circle to a chord ____.

The perpendicular drawn from the center of a circle to a chord bisects the chord. This property is useful in solving many geometric problems involving circles and chords. It helps in finding the length of the chord and other related measurements.

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In a right triangle inscribed in a circle with one side as the diameter, what is the measure of the angle opposite to the diameter?

In a right triangle inscribed in a circle with one side as the diameter, the angle opposite to the diameter is 90°. This is a consequence of the Thales’ theorem.

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The distance from a point outside the circle to the point of tangency can be found using the ____.

The Tangent-Secant Theorem states that the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external segment. This relationship helps in calculating the distance from a point outside the circle to the point of tangency.

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If two tangents are drawn from an external point to a circle, then the angle between the two tangents is equal to ____.

The angle between two tangents drawn from an external point to a circle is equal to the angle at the center subtended by the chord joining the points of tangency. This property is useful in various geometric problems and constructions involving tangents and circles.

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The length of a tangent from a point 13 units away from the center of a circle is 12 units. What is the radius of the circle?

Using the Pythagorean theorem, we get (tangent length)^2 + (radius)^2 = (distance from center)^2. So, radius = √((13)^2 – (12)^2) = √(169 – 144) = √25 = 5 units.

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A circle has a radius of 11 units. Calculate the length of a tangent from a point 15 units away from the center.

Using the Pythagorean theorem, we get (tangent length)^2 + (radius)^2 = (distance from center)^2. So, tangent length = √((15)^2 – (11)^2) = √(225 – 121) = √104 = 8 units.

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A billboard is 15 units high. If a person stands 20 units away from the base of the billboard, what is the length of the hypotenuse formed between the top of the billboard and the person?

Using the Pythagorean theorem, we get (hypotenuse)^2 = (height)^2 + (distance from base)^2. So, hypotenuse = √((15)^2 + (20)^2) = √(225 + 400) = √625 = 25 units.

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Find the length of the tangent from a point 10 units away from the center of a circle with a radius of 6 units.

Using the Pythagorean theorem, we get (tangent length)^2 + (6)^2 = (10)^2. So, tangent length = √((10)^2 – (6)^2) = √(100 – 36) = √64 = 8 units.

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The line segment that touches a circle at one endpoint and extends outside the circle is called ____.

A tangent is a line segment that touches a circle at one endpoint and extends outside the circle. It is distinct from a chord, which lies entirely inside the circle, and a secant, which intersects the circle at two points. The unique property of a tangent is that it touches the circle at exactly one point.

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A ladder 16 units long is placed against a wall such that its base is 12 units away from the wall. How high does the ladder reach on the wall?

Using the Pythagorean theorem, we get (ladder length)^2 = (height)^2 + (base distance)^2. So, height = √((16)^2 – (12)^2) = √(256 – 144) = √112 = 14 units. This calculation gives the vertical height the ladder reaches on the wall.

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A circle is described with center O and radius r. If P is an external point, and PA and PB are tangents from P to the circle, then the triangle PAB is ____.

In the triangle PAB, where PA and PB are tangents from P to the circle, the triangle is isosceles because PA = PB. This is due to the equality of tangent segments from an external point to the points of tangency.

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If two circles touch each other externally, the distance between their centers is equal to ____.

If two circles touch each other externally, the distance between their centers is equal to the sum of their radii. This is because the point of tangency lies on the line segment joining the centers of the two circles, making the distance equal to the sum of the radii.

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In a cyclic quadrilateral, the product of the lengths of its diagonals is equal to ____.

In a cyclic quadrilateral, the product of the lengths of its diagonals is equal to the product of the lengths of its opposite sides. This property is known as Ptolemy’s theorem, which is a fundamental result in circle geometry.

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A tangent to a circle forms a 30° angle with a chord that it intersects. The angle subtended by the chord at the center of the circle is ____.

When a tangent to a circle forms a 30° angle with a chord, the angle subtended by the chord at the center of the circle is twice the angle between the chord and the tangent. Therefore, it is 2 × 60° = 120°.

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A pole 12 units tall casts a shadow of 5 units. What is the distance from the top of the pole to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((12)^2 + (5)^2) = √(144 + 25) = √169 = 13 units. This represents the hypotenuse in the right triangle formed.

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If the radius of a circle is doubled, the length of the tangent from an external point to the circle ____.

If the radius of a circle is doubled, the length of the tangent from an external point to the circle remains the same. The length of the tangent depends on the distance from the external point to the center and the original radius, but not directly on the radius alone.

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The circle that passes through all the vertices of a polygon is called ____.

The circle that passes through all the vertices of a polygon is called the circumcircle. The center of this circle is known as the circumcenter, and it is equidistant from all the vertices of the polygon.

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A circle is inscribed in a triangle if ____.

A circle is inscribed in a triangle if it is tangent to each side of the triangle. This means that each side of the triangle is tangent to the circle at a single point. The points of tangency divide the sides of the triangle into segments that are used in various geometric proofs and constructions.

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The distance between the centers of two circles is 17 units. If the radii of the circles are 10 units and 7 units respectively, find the length of the common external tangent.

The length of the common external tangent can be found using the formula: √((distance between centers)^2 – (difference of radii)^2). Therefore, length of the common external tangent = √((17)^2 – (3)^2) = √(289 – 9) = √280 = 15 units.

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A vertical wall is 8 units high and casts a shadow of 6 units on the ground. What is the distance from the top of the wall to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((8)^2 + (6)^2) = √(64 + 36) = √100 = 10 units.

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The common tangent to two circles that does not intersect the line segment joining their centers is called an external common tangent. This type of tangent is outside both circles and touches them at exactly one point each.

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If two tangents are drawn to a circle from an external point, the tangents are ____.

When two tangents are drawn from an external point to a circle, they are equal in length. This property is used in many geometric proofs and problems.

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A person stands 12 units away from the base of a lighthouse. If the lighthouse is 20 units tall, what is the length of the shadow cast by the lighthouse on the ground?

Using the Pythagorean theorem, we have (height)^2 = (distance from base)^2 + (shadow length)^2. So, shadow length = √((20)^2 – (12)^2) = √(400 – 144) = √256 = 16 units.

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A tree casts a shadow of 15 units on the ground. If the tree is 9 units tall and the angle of elevation of the sun is 45 degrees, what is the distance from the top of the tree to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((9)^2 + (15)^2) = √(81 + 225) = √306 = 18 units.

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The point where a tangent touches a circle is called the point of tangency. This is the only point where the tangent and the circle intersect.

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A water tank casts a shadow of 9 units when the height of the tank is 12 units. What is the distance from the top of the tank to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((12)^2 + (9)^2) = √(144 + 81) = √225 = 15 units. This distance represents the hypotenuse of the right triangle formed.

52 / 66

A building 18 units high casts a shadow of 24 units. What is the distance from the top of the building to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((18)^2 + (24)^2) = √(324 + 576) = √900 = 30 units. The hypotenuse in this right triangle is the distance from the building top to the shadow tip.

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A circle is circumscribed about a quadrilateral if ____.

A circle can be circumscribed about a quadrilateral if and only if the sum of opposite angles of the quadrilateral is 180 degrees. This property is known as the supplementary angles property of cyclic quadrilaterals.

54 / 66

A flagpole 15 units high casts a shadow of 20 units. What is the distance from the top of the flagpole to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((15)^2 + (20)^2) = √(225 + 400) = √625 = 25 units. The hypotenuse formed in this scenario represents the direct distance from the top of the flagpole to the shadow’s tip.

55 / 66

A car tire touches the ground at a single point. This point is known as the ____.

The point where a car tire touches the ground is known as the tangent point. This is because the tire, which can be modeled as a circle, touches the ground at exactly one point. The ground acts as a tangent to the tire at this point.

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The angle between a tangent and a radius at the point of tangency is ____.

The angle between a tangent and the radius drawn to the point of tangency is always 90 degrees. This is because the tangent is perpendicular to the radius at the point of contact. This property is fundamental in many geometric proofs and constructions involving circles and tangents.

57 / 66

A building casts a shadow of 30 units on the ground. If the height of the building is 18 units, what is the distance from the top of the building to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((18)^2 + (30)^2) = √(324 + 900) = √1224 = 36 units.

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The length of the tangent from a point outside the circle to the point of tangency is called the ____.

The length of the tangent from a point outside the circle to the point of tangency is called the tangent segment. This segment is an important element in problems involving tangents and circles, as it is often equal to other tangent segments drawn from the same external point.

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A triangle inscribed in a circle such that one of its sides is the diameter of the circle is called ____.

A triangle inscribed in a circle such that one of its sides is the diameter of the circle is called a right triangle. This is a consequence of the fact that the angle subtended by the diameter at the circumference is a right angle (90°).

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In a circle, the radius to the point of tangency is always ____.

In a circle, the radius to the point of tangency is always perpendicular to the tangent. This is a fundamental property of tangents and circles. The tangent is perpendicular to the radius at the point of contact, which is a critical concept in circle geometry.

61 / 66

The distance between the centers of two circles is 20 units. If the radii of the circles are 12 units and 8 units respectively, find the length of the common external tangent.

The length of the common external tangent can be found using the formula: √((distance between centers)^2 – (difference of radii)^2). Therefore, length of the common external tangent = √((20)^2 – (4)^2) = √(400 – 16) = √384 = 16 units.

62 / 66

A ladder 20 units long is placed against a wall such that its base is 16 units away from the wall. How high does the ladder reach on the wall?

Using the Pythagorean theorem, we get (ladder length)^2 = (height)^2 + (base distance)^2. So, height = √((20)^2 – (16)^2) = √(400 – 256) = √144 = 12 units. The ladder reaches this height on the wall.

63 / 66

A lamppost 14 units tall casts a shadow of 21 units. What is the distance from the top of the lamppost to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((14)^2 + (21)^2) = √(196 + 441) = √637 = 25.24 units (approximated to 28 units). The hypotenuse is the distance from the lamppost top to the shadow tip.

64 / 66

A tree of height 10 units casts a shadow of 24 units. Find the distance from the top of the tree to the tip of the shadow.

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((10)^2 + (24)^2) = √(100 + 576) = √676 = 26 units. This distance forms the hypotenuse of the right triangle formed by the tree height and shadow length.

65 / 66

A statue is 7 units tall and casts a shadow of 24 units on the ground. What is the distance from the top of the statue to the tip of the shadow?

Using the Pythagorean theorem, we get (distance from top to tip of shadow)^2 = (height)^2 + (shadow length)^2. So, distance = √((7)^2 + (24)^2) = √(49 + 576) = √625 = 25 units.

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The length of a tangent from a point 5 units away from the center of a circle is 4 units. What is the radius of the circle?

Using the Pythagorean theorem, we get (tangent length)^2 + (radius)^2 = (distance from center)^2. So, radius = √((5)^2 – (4)^2) = √(25 – 16) = √9 = 3 units.

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