📐 geometry

1. **Problem statement:** (a)(i) Given the circle $k$ with equation $$x^2 + y^2 = 49,$$ find the centre and radius.

1. **Problem statement:** We are given a children's slide with points X, Y, Z, U, V, and W.

1. **Problem statement:** In parallelogram ABCD, given that $ED = 7x - 13$ and $BD = 16x - 38$, find the length of diagonal $BD$. 2. **Key property:** In a parallelogram, the diago

1. **State the problem:** We are given two line segments \(\overline{DE}\) and \(\overline{FG}\) intersecting at point \(H\), where \(H\) is the midpoint of both segments. We know

1. **State the problem:** We have a right triangle with angles 30°, 60°, and 90°. The side opposite 30° is given as $22\sqrt{3}$, and we need to find the lengths of sides $x$ (hypo

1. **State the problem:** We need to find the area of a right triangle where one leg is 13 units and the angle adjacent to this leg is 51°. 2. **Identify the known values:**

1. **Problem statement:** Given triangles $\triangle ABC$ and $\triangle ADE$ are similar, find $x$. 2. **Known sides:**

1. **State the problem:** We need to find the volume of a composite solid consisting of a rectangular prism base and two triangular prisms on top. 2. **Identify the shapes and dime

1. **State the problem:** Alexis measured the distance across a dinner plate as 22 cm. We need to estimate the radius, diameter, and circumference of the plate.

1. **State the problem:** We need to estimate the area of a circle with a diameter of 40 feet using $\pi \approx 3.14$. 2. **Recall the formula for the area of a circle:**

1. **Stating the problem:** We need to find the value of $x$ given the angles and relationships in the diagram. 2. **Given information:**

1. **State the problem:** We are given three angles around a point: one is $90^\circ$, another is $40^\circ$, and the third is $(6x + 2)^\circ$. We need to find the value of $x$. 2

1. **Problem statement:** In trapezoid PQST, UR is the midsegment. Given bases PQ = 11 and TS = 18, find the length of UR. 2. **Formula for midsegment of trapezoid:** The midsegmen

1. **State the problem:** Plot the point $(-6, 1)$ on a Cartesian coordinate grid. 2. **Understanding the coordinate system:** The point $(-6, 1)$ means move 6 units to the left on

1. **State the problem:** We need to find the perimeter of a figure composed of a parallelogram and a semicircle attached on the side of length 8. 2. **Identify the sides:** The pa

1. **State the problem:** We need to find the total volume of a solid made by joining a cylinder and a hemisphere, both having radius $1.5$ cm. The cylinder's height is $4$ cm. 2.

1. **Problem (d):** Show that a triangle with sides 10 cm, 5 cm, and 11 cm is not right-angled. 2. **Step 1:** Recall the Pythagorean theorem for right-angled triangles: $$a^2 + b^

1. **State the problem:** We need to find the perimeter of a figure composed of a rectangle and two semicircles attached to the vertical ends of the rectangle. 2. **Identify given

1. **State the problem:** We have an isosceles trapezoid with sides AB = 3x - 2, BC = 5x - 6, CD = 3x + 9, and AD = 4x + 5. We need to find the value of $x$ and then find $m\angle

1. **State the problem:** We have an isosceles triangle with two equal sides of 15 units each and a base of 7x units. The perimeter is given as 17x units. 2. **Write the formula fo

1. **State the problem:** We need to estimate the number of vertical cross sections required so that their total area equals the area of the base of a rectangular prism with dimens