📐 geometry

1. The problem asks for the volume of a sphere with radius $7$ inches. 2. The formula for the volume of a sphere is $$V = \frac{4}{3} \pi r^3$$ where $r$ is the radius.

1. **Problem:** Find $x$ and the missing angle when two parallel lines are cut by a transversal, with angles $100^\circ$ and $(x+10)^\circ$ on the same side exterior. 2. **Formula

1. **State the problem:** Given a right triangle $\triangle FKJ$ with right angle at vertex $K$, prove that $a^2 + e^2 = d^2$ where $a = FK$, $e = KJ$, and $d = FJ$ (the hypotenuse

1. **State the problem:** We have a right triangle with points J, M, L, and K. Line segment JM is perpendicular to ML, so angle M is 90°. Given: JM = 2, MK = 8, and we need to find

1. **State the problem:** We have two similar quadrilaterals with corresponding sides proportional. We need to find the lengths of $x$, $y$, and $z$ in the smaller quadrilateral gi

1. The problem states that triangles \(\triangle JKL\) and \(\triangle MKN\) are similar (\(\triangle JKL \sim \triangle MKN\)) and asks to find the value of \(x\). 2. In similar t

1. **State the problem:** We have a right triangle OMN with a right angle at N, angle O is 44°, side ON is 9, and side OM is labeled $x$. We need to find $x$. 2. **Identify the sid

1. **Stating the problem:** Given a polygon with points and segments where $EP = FP$, a quadrilateral $ABCD$ with $AC = BD$ and angles $\angle ACD = \angle BDC$, and a trapezoid $A

1. **State the problem:** We need to find the area of a parallelogram with sides 5 m and 4 m, and an included angle of 60 degrees. 2. **Formula for area of a parallelogram:**

1. **Problem Statement:** Find the measure of angle $\angle I$ in triangle $\triangle IJK$ where sides $i=96$ cm, $j=22$ cm, and $k=85$ cm.

1. **State the problem:** We are given a triangle ΔOPQ with sides $o=290$ inches, $p=720$ inches, and $q=600$ inches. We need to find the measure of angle $\angle P$ to the nearest

1. **State the problem:** We are given two right triangles sharing a common hypotenuse line. The smaller triangle has legs $x$ and $h$ and hypotenuse $y$. The larger triangle has l

1. **Problem statement:** Given two parallel lines $m \parallel n$ and a transversal line intersecting them, find the value of angle $x$ when the angle on the bottom line is $109^\

1. **State the problem:** We have a quadrilateral with angles at vertices F, M, S, and G. The angles at M, S, and G are given as 85°, 127°, and 62° respectively. We need to find th

1. **State the problem:** Given parallelogram ABCD with points N on AD, K on AB, M on DC, L on BC, and O as the intersection of lines NK and ML, prove that KO = OM and NO = OL.

1. **Problem Statement:** Given a rectangle DEFG with side GF = 11 and side GH = 14, find the lengths GE, DF, HF, and DG.

1. Problem: Calculate the radius of the circumscribed circle (circumradius) of a triangle with sides $4\sqrt{2}$, $8$, and $4\sqrt{2}$. 2. Formula: The circumradius $R$ of a triang

1. Stated problem: Calculate the sine of the largest angle in a triangle with sides 7, 8, and 9. 2. To find the sine of the largest angle, first identify the largest side, which is

1. **State the problem:** Calculate the area of a circle with diameter 4 cm. 2. **Recall the formula:** The area $A$ of a circle is given by

1. **State the problem:** Calculate the area of a circle with radius $r = 8$ inches. 2. **Formula:** The area $A$ of a circle is given by the formula:

1. **State the problem:** We have two similar triangles, a large right triangle VZX and a smaller triangle WYX inside it. Given lengths VW = 48, WX = 32, and YX = 30, we need to fi