📐 geometry

1. **State the problem:** We need to find the length of the hypotenuse $c$ of a right triangle where the legs are 54 m and 72 m. 2. **Formula used:** According to the Pythagorean t

1. **State the problem:** We have a right triangle with legs of lengths 60 meters and 80 meters, and we need to find the length of the hypotenuse $c$. 2. **Formula used:** Accordin

1. **State the problem:** We are given the parametric equations of an ellipse:

1. **State the problem:** Given points and angles with various congruences and perpendicularities, prove relationships between angles and triangles using geometric theorems. 2. **G

1. **State the problem:** We are given a circle with center $(-2, 7)$ and radius $\sqrt{5}$. We need to write the equation of the circle in standard form and describe its graph.

1. **State the problem:** We have a right triangle with legs of lengths $a$ and 6 yards, and a hypotenuse of 10 yards. We need to find the perimeter, which is the sum of all three

1. **State the problem:** We have a right triangle with base $12$ cm, height $16$ cm, and hypotenuse $c$. We need to find the perimeter, which is the sum of all sides. 2. **Formula

1. **State the problem:** We have a right triangle with one leg of length 9 yards, a hypotenuse of length 15 yards, and the other leg labeled as $b$. We need to find the perimeter

1. **State the problem:** We are given a circle with center $(-6, 1)$ and radius $1$. We need to write the equation of the circle in standard form and graph it.

1. **State the problem:** We need to find the center and radius of the circle given by the equation $$(x - 3)^2 + (y + 6)^2 = 4$$. 2. **Recall the formula:** The general form of a

1. **Stating the problem:** We have a triangle inscribed in a circle with angles 67°, 9°, and an unknown angle. Outside the circle, at one vertex of an extended triangle, there is

1. **State the problem:** You drive 9 miles west, then 6 miles north. We need to find the straight-line distance from the start to the end point. 2. **Formula used:** This is a rig

1. **State the problem:** A 20-foot ladder leans against a house reaching 16 feet high. Christian pulls the base 2 feet farther from the house. We need to find the new height the l

1. **State the problem:** We are given the sum of the interior angles of a polygon as 1620° and need to find the number of sides of the polygon. 2. **Formula:** The sum of the inte

1. **State the problem:** We are given five angles of a hexagon and need to find the measure of the sixth angle. 2. **Recall the formula:** The sum of the interior angles of a poly

1. **State the problem:** We have a right triangle with sides 7.7, 11, and the hypotenuse labeled as $x$ and 10 (likely a typo, we consider $x$ as the hypotenuse). 2. **Identify th

1. **State the problem:** We need to find the surface area of a triangular prism with given side lengths. 2. **Identify the dimensions:** The triangular base has sides 5 m (height)

1. **State the problem:** We are given three points: $A(-3,4)$, $B(2,1)$, and $C(-4,-5)$. We need to find a fourth point $D(x,y)$ such that $ABCD$ forms a kite. 2. **Recall kite pr

1. **Stating the problem:** We have two similar isosceles triangles ABC and DAB with AB = AC and AD = BD. We know the ratio of sides BC to CD is 4:21. We need to find the ratio AB

1. **State the problem:** Susan has a round cake with a diameter of 20 cm. She wants to put a ribbon around the cake. We need to find the least length of ribbon required. 2. **Form

1. **State the problem:** Calculate the length on a scale drawing in centimeters for an actual length of 60 meters with a scale of 1:15.