Intermediate Level Geometry Test: Quiz!

When a perpendicular is drawn from the center of a circle to a chord, it will always bisect the chord. This is because the perpendicular line will intersect the chord at its midpoint, dividing the chord into two equal segments. Therefore, the statement is true.

The statement is true because the diagonal of a parallelogram divides the parallelogram into two congruent triangles. This can be proven using the properties of parallelograms and the fact that opposite sides of a parallelogram are equal in length and parallel to each other. When the diagonal is drawn, it intersects the parallelogram at two points, dividing it into two triangles. Since the opposite sides of the parallelogram are equal, the triangles formed by the diagonal are congruent.

In a circle, the distance from the center to any point on the circle is the same. Since equal chords are chords that have the same length, they must be equidistant from the center. This is because if two chords have the same length, their endpoints are at the same distance from the center, and hence they are equidistant. Therefore, the statement "Equal chords are equidistant from the center of a circle" is true.

A diameter is a line segment that joins two points on the circumference of a circle and passes through the center of the circle. It is the longest chord in a circle and divides the circle into two equal halves.

In a circle, chords that are equidistant from the center are equal in length. Therefore, the statement that the nearer chord from the center is shorter than the farther chord is false.

If the altitudes drawn from every vertex of a triangle are equal, it means that the perpendicular distances from each vertex to the opposite side are the same. This can only happen in an equilateral triangle, where all three sides and angles are equal. In an isosceles triangle, only two sides and angles are equal, while in a right-angled triangle, one angle is equal to 90 degrees. A scalene triangle has no equal sides or angles. Therefore, the correct answer is an equilateral triangle.

The distance of a chord from the center of a circle is equal to the perpendicular distance from the center to the chord. In this case, the chord is 24 cm long and the radius of the circle is 20 cm. According to the properties of a circle, the perpendicular distance from the center to a chord bisects the chord. Therefore, the distance from the center to the chord is half the length of the chord, which is 12 cm.

Congruent triangles have the same shape and size, meaning all corresponding sides and angles are equal. Since the area of a triangle is determined by its base and height, and congruent triangles have equal corresponding sides, the base and height of the triangles will also be equal. Therefore, the area of congruent triangles will be the same. Hence, the statement is true.

A chord is a line segment that connects two points on a circle. In this question, a 12 cm long chord is drawn at a distance of 8 km from the center of the circle. To find the radius of the circle, we can use the formula for the distance between the center of a circle and a chord, which is r^2 = d^2 + (c/2)^2, where r is the radius, d is the distance between the center and the chord, and c is the length of the chord. Plugging in the given values, we get r^2 = (8 km)^2 + (12 cm/2)^2. Simplifying this equation gives us r^2 = 64 km^2 + 36 cm^2, which can be further simplified to r^2 = 64000000 cm^2 + 36 cm^2. Taking the square root of both sides, we find that r is approximately 100 cm or 1 m. Therefore, the radius of the circle is 10 cm.

This statement is false. While every pair of congruent triangles are also similar to each other, the reverse is not always true. Similar triangles have proportional sides and congruent angles, but they may not have the same shape or size. Congruent triangles, on the other hand, have exactly the same shape and size. Therefore, not all similar triangles are congruent to each other.

The distance of a chord from the center of a circle is equal to half the length of the perpendicular segment drawn from the center to the chord. In this case, the diameter of the circle is 10 cm, so the radius (which is also the distance from the center to any point on the circle) is 5 cm. The chord is 8 cm long, so the perpendicular segment from the center to the chord will bisect the chord into two segments, each measuring 4 cm. Therefore, the distance of the chord from the center is 4 cm, which is half of the length of the chord.

Congruent triangles have the same shape and size, which means that their corresponding sides and angles are equal. Since the sides and angles are equal, the areas of congruent triangles are also equal. Therefore, every pair of congruent triangles is equal in area. Additionally, since congruent triangles have the same shape, they are also similar. Similarity implies that the corresponding angles of the triangles are equal, and the corresponding sides are in proportion. Hence, every pair of congruent triangles is similar as well.

The distance of a chord drawn in a circle is equal to the perpendicular distance from the center of the circle to the chord. In this case, the radius of the circle is 17 cm, and the chord is 16 cm long. Since the chord is shorter than the diameter of the circle, it must be closer to the center than the radius. Therefore, the distance from the center to the chord is less than 17 cm, and since it is given as 15 cm, the statement is true.

The statement is comparing the length of the chord and the radius of a circle. It states that the length of the chord is shorter than the radius. This is a mathematical fact because the radius is the distance from the center of the circle to any point on its circumference, while the chord is a line segment that connects two points on the circumference. Since the chord is a line segment within the circle, it is always shorter than the radius, which extends from the center to the circumference.

The given answer options are all conditions for two triangles to be congruent. SSS stands for Side-Side-Side, which means that all three sides of the triangles are equal. RHS stands for Right Angle-Hypotenuse-Side, which means that one triangle has a right angle, and the hypotenuse and one side of the triangles are equal. ASA stands for Angle-Side-Angle, which means that two angles and the included side of the triangles are equal. AAS stands for Angle-Angle-Side, which means that two angles and a non-included side of the triangles are equal. SAS stands for Side-Angle-Side, which means that two sides and the included angle of the triangles are equal.

The given statement states that the diagonals of a quadrilateral divide it into four congruent triangles. A square and a rhombus are the only quadrilaterals that have congruent diagonals. Therefore, the quadrilateral can be either a square or a rhombus.

The line segment which joins the midpoint of two sides of the triangle is half of the third side and parallel to it. This is because the midpoint of a side divides the side into two equal parts, and when a line segment is drawn from the midpoint of one side to the midpoint of another side, it bisects the third side. Additionally, the line segment connecting the midpoints is parallel to the third side, as it can be proven using the midpoint theorem.