📐 geometry

1. **State the problem:** Given two congruent triangles $\triangle ABC \cong \triangle FDE$, find the values of $x$ and $y$ given the angles: - $\angle C = 108^\circ$

1. **Problem Statement:** Calculate the volume and surface area of a composite shape consisting of a rectangular prism and a cone attached to its side.

1. **State the problem:** We are given two similar triangles \(\triangle JKL \sim \triangle MKN\) and need to find the value of \(x\). The sides given are \(KN = 12\), \(NL = 20\),

1. **State the problem:** Given that triangles $\triangle DGH$ and $\triangle DEF$ are similar, find the value of $x$. 2. **Recall the similarity rule:** Corresponding sides of sim

1. **State the problem:** Given that triangles $\triangle JKL$ and $\triangle NMP$ are similar, find the value of $x$. 2. **Recall the property of similar triangles:** Correspondin

1. **Problem statement:** We have a circle with center P and two radii each of length $x$ forming a right angle. A tangent line touches the circle at the endpoint of one radius, cr

1. **Problem statement:** Given a circle with two tangent lines meeting outside the circle, the radius to the tangent points is $x$, and the base between tangent points is 6. Also,

1. **Problem statement:** Find the coordinates of the circumcenter of triangle J(5,0), K(5,-8), and L(0,0). Also find the equations of two perpendicular bisectors and their interse

1. **State the problem:** We need to find the length of leg $a$ in a right triangle where the hypotenuse is 12 and the other leg is 6. 2. **Formula used:** In a right triangle, the

1. **Stating the problem:** We have two similar triangles, $\triangle JKL \sim \triangle QRS$. Given sides of $\triangle QRS$ are $QR=65$ and $RS=45$. For $\triangle JKL$, side $LK

1. **State the problem:** We are given that the ratio of one side of triangle ABC to the corresponding side of triangle DEF is $5:8$, and the perimeter of triangle DEF is 96 inches

1. **State the problem:** We need to find the area of a walkway formed by four semicircles. The inner semicircles have a diameter of 14 metres, and the walkway width between the in

1. **Problem statement:** We need to find the side length of the smallest square plate on which a 22-cm chopstick can fit along the diagonal without any overhang. 2. **Formula used

1. **Problem Statement:** Prove the SAS (Side-Angle-Side) congruence theorem which states that if two sides and the included angle of one triangle are equal to two sides and the in

1. **Problem statement:** Given pentagon ABCDE with circle center O passing through A, C, D, E, angles $\angle EAC=36^\circ$, $\angle CAB=78^\circ$, and $AB \parallel DC$. Find $x,

1. **State the problem:** Prove that two triangles are congruent if two sides and the included angle are the same. 2. **Recall the SAS Congruence Theorem:** If two sides and the in

1. **Problem statement:** Given pentagon ABCDE with circle center O passing through A, C, D, E, angles $\angle EAC=36^\circ$, $\angle CAB=78^\circ$, and $AB \parallel DC$. Find $x$

1. ප්‍රශ්නය: වෘත්තයක් තුළ U මධ්‍යස්ථානය වන අතර P, Q, R, S, T, U, V යන ලක්ෂ්‍ය සහ PS, PR, QR, QT, TR, UT යන රේඛා කොටස් ඇත. QT = TR සහ QTU = UTR යන සමානතා දී ඇත. S Q V + U V R = S P

1. Problem statement. In the cyclic pentagon ABCDE the points A, C, D and E lie on a circle with centre O, AB is parallel to DC, $\angle EAC = 36^\circ$ and $\angle CAB = 78^\circ$

1. **Problem statement:** Calculate the area of the rectangle with given dimensions 6 m (top line) and 4 cm (right line) using the scale 1 : 40,000. 2. **Understanding the scale:**

1. **State the problem:** We have a circle with points A, B, C, D on the circumference and a tangent EBF at point B. Given angles are $\angle BAC = 40^\circ$ and the angle between