📐 geometry

1. **Problem Statement:** Quadrilateral MNOP is dilated by a scale factor of $\frac{3}{4}$ to form quadrilateral M'N'O'P'. We need to find the length of side O'P'. 2. **Given:** Si

1. The problem asks us to find the scale factor between two squares, where the left square has side length 9 and the right square has side length 40. 2. The scale factor is the rat

1. **State the problem:** We have two squares, one with side length 18 units and the other with side length 66 units. We need to find the scale factor from the smaller square to th

1. **Stating the problem:** We have two squares, one smaller with side length $18$ and one larger with side length $66$. We want to find properties such as area or perimeter of the

1. **State the problem:** We have two right triangles, one larger with legs 16 and 16, and a smaller one with legs 9 and 9. We need to find the scale factor from the larger triangl

1. **Problem statement:** We have a large square with side length 9 cm divided into four smaller rectangles by one vertical and one horizontal line. The top-right rectangle has dim

1. **Problem Statement:** We have a triangle in the lower-left quadrant with vertices approximately at $(-1,-1)$, $(-5,-5)$, and $(-5,-1)$. We want to find the image of this triang

1. **Problem statement:** We are given a rhombus ABCD with diagonal AC = 13 units. Point E is the intersection of the diagonals, and it forms a right angle with AC. The diagonal BC

1. **State the problem:** We have an isosceles triangle ABC with angles at B and C given. Angle at B is $2x^2$ degrees, and angle at C on the straight line CD is $115^\circ$. We ne

1. **State the problem:** We have a right triangle JIH with a right angle at vertex I. Given:

1. **State the problem:** We need to find the value of $x$ such that point $N$ is the incenter of triangle $ABC$. The incenter is the point where the angle bisectors of the triangl

1. **State the problem:** We have an isosceles triangle with two equal angles each measuring $58.4^\circ$ and a perpendicular height of $16.3$ cm. We need to find the length of the

1. **State the problem:** We have triangle ABC with point D on AC such that angle BDC = 90°.

1. **Problem Statement:** Find the area and perimeter of an L-shaped figure composed of five equal squares arranged with three squares vertically on the left and two squares horizo

1. **State the problem:** We are given a line with equation $y - 2x = 7$ which is the perpendicular bisector of the line segment $AB$ where $A = (j, 7)$ and $B = (6, k)$. We need t

1. **State the problem:** Determine if the given triangles are right triangles based on their side lengths. 2. **Recall the Pythagorean theorem:** For a triangle with sides $a$, $b

1. **Problem:** Find the length of diagonal QS in rectangle PQRS given that diagonal PR = 12. 2. **Recall:** In a rectangle, the diagonals are congruent. This means $$PR = QS$$.

1. The problem involves finding the value of the angle $x$ in a circle where two chords intersect, creating angles of $68^\circ$ and $x^\circ$. 2. The key rule for angles formed by

1. **State the problem:** We have a classroom floor and a closet floor combined with a total area of 71 square meters. The classroom is a rectangle with dimensions 7 m by 9 m. We n

1. **State the problem:** We are given triangle $\triangle ABC$ with midsegments $DE$, $EF$, and $DF$. We need to find the coordinates of points $D$, $E$, and $F$ (the midpoints of

1. **Problem statement:** In a right-angled triangle with legs $x$ and $y$, one leg is extended by 4 cm and the other shortened by 5 cm, resulting in an isosceles right-angled tria