📐 geometry

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

1. **State the problem:** We need to find the hypotenuse of a right triangle when the other two sides are 25 and 95. 2. **Formula used:** According to the Pythagorean theorem, the

1. **State the problem:** We have a closed cylindrical hot water tank with volume 300 litres. We want to find the height $h$ in terms of $\pi$ and $r$, and then show the total surf

1. **Problem statement:** Determine the effect on volume when dimensions of cubes, cylinders, and rectangular prisms are changed as described. 2. **Formulas:**

1. **Problem 1: Calculate the value of the pronumeral $x$ using Pythagoras' Theorem.** 2. Pythagoras' Theorem states that in a right triangle, the square of the hypotenuse ($c$) eq

1. **State the problem:** We have a right triangle with a base segment of 11.4 cm, an extended base segment of 7.1 cm, a hypotenuse of 12.2 cm, and an unknown height $x$. We need t

1. The problem is to find the formula for the surface area of a sphere. 2. The formula for the surface area $A$ of a sphere with radius $r$ is: