📐 geometry

1. **Problem statement:** We have points O(0,0), A(8,6), and P(x,10) where P moves along the line $y=10$. We want to show that $$\cos \angle OPA = \frac{x^2 - 8x + 40}{\sqrt{(x^2 -

1. **State the problem:** We need to solve for $x$ given the equation involving angles $m\angle DBA = (4x + 2)^\circ$ and $m\angle ABC = (5x - 9)^\circ$ that sum to $m\angle ABD =

1. **State the problem:** We are given that $m\angle ABD = 78^\circ$, and the angle $\angle ABD$ is split into two angles: $m\angle DBA = (4x + 2)^\circ$ and $m\angle ABC = (5x - 9

1. The user asks to draw diagrams for multiple problems, but as a text-based assistant, I cannot draw images directly. 2. However, I can describe how to sketch each diagram:

1. **Problem:** Determine the measure of angle DC in quadrilateral ABCD given AB = 4, AD = 4, \(\angle B = 57^\circ\), and \(\angle D = 53^\circ\). 2. **Given:** Quadrilateral ABCD

1. **Problem Statement:** Determine which transformations map an equilateral triangle exactly onto itself given lines m and n and center K.

1. **Problem statement:** We have a solid hemisphere with center $A$ at coordinates $(5,-2,-3)$ and a point $B$ on the curved surface at $(3,-6,-9)$.

1. **State the problem:** We need to find the surface area of a triangular prism with given dimensions: triangle sides 10 in., 11 in., and 11 in., and rectangular faces with dimens

1. The problem involves finding the length of diagonal AC in a quadrilateral where the diagonal is divided into two segments AE and EC. 2. Given: AC = 4x + 10, AE = 3x + y, EC = 2x

1. **State the problem:** We have a rectangle with length 9 inches and width 17 inches. We want to find the dimensions of a similar rectangle scaled by 70%. 2. **Formula and explan

1. **State the problem:** We have a blueprint with a scale factor of 1 in : 35 in, and a wall on the blueprint is 3 inches long. We want to know if the actual wall is long enough t

1. **Stating the problem:** We have a parallelogram with angles labeled $y^\circ$, $3y^\circ$, and $3x^\circ$. We want to find the values of $x$ and $y$. 2. **Recall properties of

1. **Problem statement:** In parallelogram ABCD, points E and F lie on diagonal BD such that AE and CF are altitudes in triangles BAD and BCD, respectively. Show that $AF = CE$. 2.

1. **Problem Statement:** Find the value of $x$ in a triangle where a smaller triangle is formed by connecting the midpoints of the sides of a larger triangle. The larger triangle

1. **Problem statement:** We need to find the measure of angle $x$ formed on the circumference by two chords intersecting inside a circle. 2. **Relevant theorem:** The angle formed

1. **Problem Statement:** Given two triangles \(\triangle KLM\) and \(\triangle JMN\) with sides \(KL=6\), \(LM=10\), \(JK=3\), and \(MN=5\), determine if they are similar and stat

1. **State the problem:** We are given two spheres with radii 11 inches and 5 meters, and we need to find their surface areas and volumes.

1. The problem involves understanding the reflection of a quadrilateral across the vertical line $x = -2$. 2. The line of reflection is given by $x = -2$. This means every point on

1. **Problem statement:** We are given four lines with two pairs of parallel lines: $p \parallel t$ and $r \parallel s$. There is an angle labeled $130^\circ$ and another angle lab

1. **State the problem:** Find the volume of a triangular prism with a right triangle base where the base height is 5 cm, base length is 6 cm, and the prism length (depth) is 10 cm

1. **Problem Statement:** Find the volume of the triangular prism with a triangular base area of 15 cm² and a prism length of 5 cm. 2. **Formula for Volume of a Triangular Prism:**