📐 geometry

1. **State the problem:** Find the perimeter of each polygon given the side lengths. 2. **Recall the perimeter formula:** The perimeter $P$ of a polygon is the sum of the lengths o

1. **State the problem:** We have a right triangular prism with a base triangle having legs 5 in and 9 in, and height 12 in. We need to find: (a) The length $a$, which is the hypot

1. **State the problem:** We have a convex pentagon with interior angles 82°, 121°, 129°, 147°, and an unknown angle $x$. We need to find $x$. 2. **Formula used:** The sum of inter

1. The problem is to find the angle corresponding to the second half of 2 in a triangle context. 2. Assuming the problem refers to dividing an angle of 2 radians into two equal par

1. **Problem 19:** Find the area of the shaded region inside a circle with radius 6. 2. The formula for the area of a circle is $$A = \pi r^2$$ where $r$ is the radius.

1. **State the problem:** We need to find the measure of angle $\angle MPQ$ given two expressions for angles formed by a transversal intersecting two parallel lines. 2. **Identify

1. **State the problem:** We are given two angles formed by a transversal intersecting two parallel lines. The angles are \( (12x - 23)^\circ \) and \( (71 - x)^\circ \). We need t

1. The problem asks which quadrilateral has at least one pair of perpendicular sides. 2. Perpendicular sides meet at a right angle, which is $90^\circ$.

1. The problem asks: Which quadrilateral appears to have 4 right angles? 2. A quadrilateral is a polygon with 4 sides. A right angle is an angle of exactly 90 degrees.

1. **State the problem:** Find the volume of a cylinder with radius $r=10$ cm and height $h=20$ cm. 2. **Formula:** The volume $V$ of a cylinder is given by the formula:

1. **State the problem:** We need to find the volume of a cylinder with height $h=12$ feet and radius $r=4$ feet. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** We need to reflect given points over the line $y = -1$. 2. **Formula and rule:** To reflect a point $(x,y)$ over the horizontal line $y = k$, the reflecte

1. **Problem statement:** Reflect points across the line $y = -x$ and then across the line $y = -1$. Given points $X(-3,-1)$, $Y(-1,1)$, and $Z(3,2)$, find the coordinates of $X''$

1. **State the problem:** Find the volume of a stepped rectangular prism with given dimensions. 2. **Understand the shape:** The solid can be divided into three rectangular prisms

1. **State the problem:** We have a right triangle with one leg of length 40, an angle adjacent to this leg of 22°, and the hypotenuse labeled as $x$. We want to find the length of

1. **Problem statement:** Given quadrilateral with vertices U, T, X, V and point W on segment VX, where UV is parallel to TW. Given lengths: $UT=12$, $TX=24$, $UV=22$, and we need

1. **State the problem:** We have two triangles, VWY and UYX, with VW parallel to UX. Given lengths UV = 10, UY = 20, and YX = 44, we need to find WX. 2. **Use the properties of pa

1. The problem is to verify if the given triangle with sides $x = 8\sqrt{3}$, $y = 16\sqrt{3}$, and the side opposite the 30° angle being 8, forms a right-angled triangle with the

1. **Stating the problem:** We have a right triangle divided into two right-angled triangles by a height $a=2$ perpendicular to the base $h$. One segment of the base is $s_1=1$, an

1. **State the problem:** We need to find the value of $x$ using geometric means from the given right triangles with sides labeled $x$, $x+3$, and segments of length $x$ and $12-x$

1. **State the problem:** We have pairs of right triangles with angles labeled as $(y + 4)^\circ$ and $(2y - 15)^\circ$ and sides involving expressions with $x$. We want to find th