📐 geometry

1. **Problem Statement:** We need to find how many triangles can be formed by drawing diagonals from a common vertex in a regular octagon (8-sided polygon). 2. **Formula and Explan

1. **Problem:** Find the missing side lengths of a right triangle with a 45° angle, hypotenuse 6, opposite side $v$, adjacent side $u$. 2. **Formula:** In a 45°-45°-90° triangle, t

1. **Problem:** Find $x$ in a right triangle with legs 4 and $x$, hypotenuse 10. 2. **Formula:** Use the Pythagorean theorem: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse.

1. **Stating the problem:** We are given a circle with points A, B, C, D, E on the circumference. We know $m\angle ACB = 66^\circ$ and the length $mAB = 132$. We need to find $m\an

1. **Problem statement:** Given a circle with center C, points Q and S on the circle, and points R and T outside the circle. Segments QR and SR are tangent to the circle at Q and S

1. **Stating the problem:** We have a right triangle with hypotenuse labeled $x$, one leg labeled $7\sqrt{33}$, and a perpendicular segment from the right angle vertex to the hypot

1. **State the problem:** Verify the area calculation of the irregular polygon. 2. **Recall previous result:** The total area was calculated as $46$ km$^2$ by summing areas of two

1. **State the problem:** We need to find the area of an irregular polygon with given side lengths. 2. **Approach:** We can divide the polygon into simpler shapes (rectangles and t

1. **Problem statement:** Calculate the surface area of a cone using the formula $$\text{Surface Area} = \pi r l + B$$ where $r$ is the radius, $l$ is the slant height, and $B$ is

1. **Problem statement:** Find $x$, $y$, and $z$ in the given figures. 2. **Step 1: Triangle with hypotenuse 5, legs 3 and $x$**

1. **Problem statement:** Find $x$ and $y$ in the two right triangles where the first triangle has sides 5 (hypotenuse), 3 (one leg), and $x$ (unknown leg), and the second triangle

1. **State the problem:** Calculate the volume of a cuboid with given dimensions. 2. **Given dimensions:**

1. **Problem statement:** We need to find the angle $\theta$ between the base and the slant diagonal along the width and height of a cuboid with dimensions 17 mm (length), 10 mm (w

1. **State the problem:** Calculate the hypotenuse lengths of right-angled triangles using the Pythagorean theorem and complete the table with opposite, adjacent, and hypotenuse si

1. **Problem statement:** Find the missing side $x$ in the right triangle with legs $\sqrt{2}$ and $x$, and hypotenuse $\sqrt{10}$. 2. **Formula used:** In a right triangle, by the

1. The problem is to find the value of $x$ in a right triangle where one leg is 9, the other leg is $x$, and the hypotenuse is 12. 2. We use the Pythagorean theorem for right trian

1. The problem states that triangle ABC was dilated by 50%, creating triangle A'B'C'. 2. Dilation by 50% means the new triangle's sides are 50% of the original triangle's sides.

1. **State the problem:** We have a triangle with vertices A(1,1), B(4,1), and C(1,5). We want to find the area of the triangle after it is dilated by a scale factor of $\frac{1}{2

1. **State the problem:** Rachel wants to find the length of the lake by proving two triangles are similar and then using proportions. 2. **Identify given sides:** Left triangle si

1. **Problem statement:** We have a rectangular box with dimensions 9 cm (length), 6 cm (width), and 6 cm (height). The lid is open at an angle of 30°.

1. **Stating the problem:** We have a quadrilateral ABCD with angles $x^\circ$ at A, $y^\circ$ at D, $38^\circ$ at C, and a right angle (90$^\circ$) at B. Sides AD and DC are equal