Geometry

This page is part of the section “BSE. Geometry”. Here are solutions to typical geometry problems: area of a rectangle, inscribed angle, Pythagorean theorem, area of a trapezoid. All problems are written independently, with neat integer answers.

Problems with solutions

Problem 1. Area of a rectangle

Find the area of a rectangle with sides 6 and 4.

The area of a rectangle equals the product of its adjacent sides:

S = a \cdot b = 6 \cdot 4 = 24

Answer: 24.

Problem 2. Inscribed angle

Find the inscribed angle subtending an arc of 80°.

An inscribed angle equals half the arc it subtends:

∠ ABC = \frac{1}{2} \cdot 80 ° = 40 °

Answer: 40°.

Problem 3. Pythagorean theorem

In a right triangle the legs are 5 and 12. Find the hypotenuse.

By the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the legs:

c^{2} = a^{2} + b^{2} = 5^{2} + 12^{2} = 25 + 144 = 169

c = \sqrt{169} = 13

Answer: 13.

Problem 4. Area of a trapezoid

Find the area of a trapezoid if its bases are 4 and 8 and its height is 5.

The area of a trapezoid equals half the sum of the bases multiplied by the height:

S = \frac{a + b}{2} \cdot h = \frac{4 + 8}{2} \cdot 5 = 6 \cdot 5 = 30

Answer: 30.

Frequently asked questions

How to find the area of a rectangle?

The area of a rectangle equals the product of two adjacent sides: S = a·b. For example, for sides 6 and 4: S = 6·4 = 24.

What is an inscribed angle equal to?

An inscribed angle equals half the arc it subtends. If the arc is 80°, the inscribed angle is 40°.

How to apply the Pythagorean theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². For legs 5 and 12: c² = 25 + 144 = 169, c = 13.

How to find the area of a trapezoid?

The area of a trapezoid equals half the sum of the bases multiplied by the height: S = (a + b)/2 · h. For bases 4 and 8 and height 5: S = 6·5 = 30.