📐 geometry

1. **Problem Statement:** Find the scale factor of the sides for the similar triangles \(\triangle KLI \sim \triangle TUV\).

1. **State the problem:** We are given two parallel lines cut by a transversal, creating segments of lengths 14 and 8 on one side, and 12 and an unknown length $x$ on the other sid

1. **Problem statement:** (i) Starting from a right triangle with sides 1, 1, and $\sqrt{2}$, construct a segment of length $\sqrt{3}$ using straightedge and compass.

1. **Problem Statement:** We have a sequence of figures made by joining polygons with sides of unit length. The first three figures have perimeters 5, 8, and 11 respectively. We ne

1. The problem involves finding the area of a parallelogram with given side lengths and height. 2. The formula for the area of a parallelogram is:

1. **Problem Statement:** (i) Starting from a right triangle with sides 1, 1, and hypotenuse $\sqrt{2}$, construct $\sqrt{3}$ using straightedge and compass.

1. **Problem statement:** Given triangle ABC with points P on AB and Q on AC such that PQ is parallel to BC, prove that $$\frac{|AP|}{|AB|} = \frac{|AQ|}{|AC|} = \frac{|PQ|}{|BC|}.

1. **Problem statement:** (i) Show that if $a,b,c > 0$ and $a^2 + b^2 = c^2$, then $a + b > c$ so that $a,b,c$ can form a triangle.

1. **Problem statement:** (i) Show that if $a, b, c > 0$ and $a^2 + b^2 = c^2$, then $a + b > c$ so that $a, b, c$ can form a triangle.

1. **Problem statement:** We have a circle with center O and a tangent line EDC touching the circle at point D. We need to find the size of angle $z$ formed between the tangent lin

1. **State the problem:** We have a circle with center $O$, a tangent line $ABC$ touching the circle at $B$, and chords $AD$, $BE$, and $CE$. We need to find the size of angle $\th

1. **Problem Statement:** We need to find a formula for the area of trapezoid ABCD in terms of $a$, $b$, and $h$. The trapezoid has one pair of parallel sides: $AB$ and $DC$. Side

1. **State the problem:** We are given a sphere with a radius of 10 meters and asked to find its volume in cubic meters. 2. **Formula for the volume of a sphere:** The volume $V$ o

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