📐 geometry

1. The problem is to state and explain the Pythagorean theorem. 2. The Pythagorean theorem applies to right-angled triangles and states that the square of the length of the hypoten

1. **State the problem:** We need to prove that triangle ABC is similar to triangle PQC. 2. **Identify corresponding angles or sides:** To prove similarity, we can use Angle-Angle

1. **State the problem:** We have a quadrilateral with points A, P, Q, C, and B. Given lengths are AP = 6 cm, PQ = $x$, QC = 12 cm, PC = 18 cm, BC = 20 cm, and AB = $y$. We want to

1. **Problem statement:** Find the length $y$ in each figure based on the given dimensions and geometric properties. ---

1. Problem: Find the angle of rotation about the origin that maps point A(4, 5) to A'(-4, -5). 2. Explanation: Rotation about the origin by angle $\theta$ transforms point $(x, y)$

1. Problem: Identify each 3D shape based on the given number of faces, edges, corners, and face types. 2. (a) The shape has 5 faces, 5 corners, and 8 edges. One face is a square, a

1. **State the problem:** We need to find the area of trapezium ABDE. 2. **Identify the dimensions:**

1. **State the problem:** We have a frustum formed by removing a small cone from a large cone. The large cone has slant height $l_1 = 12$ cm and base radius $r_1 = 6$ cm. The small

1. પ્રશ્ન: વ્યાસ 16.8 cm ધરાવતા ગોળકનું ઘનફળ શોધો. 2. ગોળકનું ઘનફળનું સૂત્ર છે: $$V = \frac{4}{3} \pi r^3$$ જ્યાં $r$ ગોળકનો અર્ધવ્યાસ છે.

1. Find the volume of Figure 1, a rectangular prism with a cylindrical hole. 2. Find the volume of Figure 2, a stepped figure made of two rectangular prisms.

1. **Find the volume of Figure 1** which is a rectangular prism with a cylindrical hole. Step 1: Calculate the volume of the rectangular prism.

1. Problem: Find the perimeter and area of each figure where circular portions are semicircles. (a) Semicircle with diameter 28 cm.

1. Problem 4(a): Find the perimeter and area of a semicircle with diameter 28 cm. - Radius $r = \frac{28}{2} = 14$ cm.

1. Problem: Find the perimeter and area of each figure where circular portions are semicircles, all dimensions in cm. (a) Semicircle with diameter 28 cm and a horizontal line below

1. **State the problem:** Calculate the volume of a cone with radius $r=18$ cm and height $h=6$ cm using the formula for the volume of a cone: $$V = \frac{1}{3} \pi r^2 h$$

1. **State the problem:** We need to calculate the cross-sectional area and the volume of a cylinder with radius $r=15$ cm and height $h=17$ cm.

1. The problem asks for the area of the shaded face of a cylinder, which is the top circular face. 2. The diameter of the circle is given as 26 mm.

1. The problem asks for the area of the shaded face of a cylinder, which is the top circular face. 2. The diameter of the circle is given as 22 mm.

1. **Problem statement:** We have a ship journey starting from Island C, traveling North East, then West, then South East, and returning to Island C.

1. **State the problem:** We have a wall that is 20 meters wide and 15 meters high, painted blue. We need to find:

1. **Problem 1: Calculate the value of p in the right triangle with angle 60° and adjacent side 10 cm, given Tan 60° = 1.7.** 2. Recall the tangent definition: $$\tan(\theta) = \fr