📐 geometry

1. **State the problem:** We have a right triangle with one leg of length 7, an angle of 45° opposite side $a$, and hypotenuse $b$. We need to find the length of side $b$. 2. **Rec

1. **Problem Statement:** We are given a right triangle with one leg of length $x$, another leg of length 15, and the hypotenuse of length 17. We need to find the value of $x$. 2.

1. **State the problem:** We need to find the value of $x$ in a right triangle where the sides are 17, 20, and $x$, with the right angle between the sides 17 and $x$. 2. **Formula

1. **State the problem:** We need to find the length of the hypotenuse $x$ in a right triangle where the legs are 15 and 11. 2. **Formula used:** According to the Pythagorean theor

1. **State the problem:** We need to find the length of the hypotenuse $x$ in a right triangle where the legs are 6 and 10. 2. **Formula used:** According to the Pythagorean theore

1. **State the problem:** We need to solve for $b$ in a right triangle where one leg is 6, the other leg is $b$, and the hypotenuse is 11. 2. **Formula used:** The Pythagorean Theo

1. **Stating the problem:** We are given a right trapezoidal prism with dimensions 8 cm (length), 4 cm (height of trapezoid), and 5 cm (base of trapezoid). We need to find the volu

1. **Problem statement:** Calculate the total volume of a triangular prism with a base triangle height of 35 cm and prism length of 120 cm. 2. **Formula:** The volume $V$ of a pris

1. **Problem statement:** Given that line \(\overleftrightarrow{AC}\) is tangent to circle \(O\) at point \(C\), and \(\angle ACB = 28^\circ\), find the measure of \(\angle ACO\).

1. **Problem statement:** We have a circle with center $O$ and radius $OC=8$ units. The line segment $AC$ is tangent to the circle at point $C$, and $AC=15$ units. Points $A$, $B$,

1. **State the problem:** We need to find the area of a blue irregular quadrilateral with given side lengths and heights, where the base is 3 cm, a vertical height from the base to

1. The problem involves finding the measures of arcs and angles related to central angles in circles. 2. Recall that a central angle in a circle is an angle whose vertex is the cen

1. Problem: Calculate the surface area of pyramids. For a square-based pyramid:

1. **Problem statement:** Given a large circle with diameter 12 cm and a shaded annulus area of 50 cm², find the diameter of the smaller inner circle.

1. **Problem statement:** Given a logo composed of three circles each with radius $6$ cm intersecting to form a shaded equilateral triangle inside, find:

1. **Problem statement:** Given three circles I, II, and III with points T (tangency of I and III), P and Q (intersections of I and II), and PR, RS, RU lengths, find the relative p

1. **Problem statement:** We have a circle with center C and radius 6 cm. We need to find the length of the arc ADB corresponding to the central angle \(\angle CAB = \frac{\pi}{8}\

1. **Stating the problem:** We have two parallel lines cut by a transversal, creating several angles. We are asked to identify which two angles are corresponding and which two are

1. **Problem statement:** Find the area of the pathway around the rectangular pool and the area of the shaded triangle. 2. **Area of the pathway:**

1. The problem asks to identify the type of symmetry that involves a rotation isometry around a center point combined with mirror lines through the center point. 2. The key terms a

1. **Stating the problem:** We are given that \(\angle AOB = 102^\circ\), \(\angle DOE\) is 11° larger than \(\angle BOC\), and lines AC and BE are straight lines. We need to find