📐 geometry

1. **Problem Statement:** Identify the shape of the cross section formed when a plane slices through each given solid.

1. **State the problem:** We need to find the surface area of a gas tank shaped as a cylinder with two hemispheres attached at each end. 2. **Given:**

1. **State the problem:** We need to find the total surface area of a shape made by joining a hemisphere on top of a cylinder. The hemisphere has radius $r=9$ cm and the cylinder h

1. **State the problem:** We need to find the total surface area of a shape made by joining a hemisphere on top of a cylinder.

1. **State the problem:** We need to prove that triangle ABC is a right triangle given that line segment BE is perpendicular to DE and that the measure of angle ABE equals the meas

1. **State the problem:** We are given two parallel lines \(\overleftrightarrow{PQ} \parallel \overleftrightarrow{RS}\) and a transversal \(\overleftrightarrow{TU}\) intersecting t

1. **State the problem:** Given that ABCD is a rhombus, prove that triangles $\triangle AEB$ and $\triangle CEB$ are congruent. 2. **Step 1:** Statement: ABCD is a rhombus. Reason:

1. **State the problem:** Given that \(\overline{BE} \cong \overline{FD}\), \(\overline{AE} \cong \overline{FC}\), and both \(\angle AEB\) and \(\angle CFD\) are right angles, prov

1. **State the problem:** Given that AECF is a parallelogram, triangles ABC and ADC are congruent, and segments EB and FD are congruent, prove that AECF is a rhombus. 2. **Recall d

1. **State the problem:** We are given a right triangle with sides $a$, $b$, and hypotenuse $c$. The Pythagorean theorem states that $$a^2 + b^2 = c^2.$$ We know $c = 5$ and $a^2 =

1. **State the problem:** We need to find the volume of a cone with height $h=19$ yards and radius $r=17$ yards. 2. **Formula for the volume of a cone:**

1. **Stating the problem:** We have a triangle ABC with right angles at B and D. Segment BD is perpendicular to AC, and AB is perpendicular to BD. Angle BDA is a right angle, and a

1. **Problem Statement:** Find the measures of angles $\angle CBD$, $\angle ABD$, and $\angle A$ in triangle $ABC$ with given conditions: $AD \perp BC$ at $D$, $BD \perp AC$ at $D$

1. **State the problem:** We have triangle KLM with angles \(\angle L = 65^\circ\), \(\angle K = 0^\circ\) (which is unusual), and an exterior angle \(\angle LMS = 115^\circ\). We

1. **Problem Statement:** We have an isosceles triangle KLM with sides KL = KM. Points L, M, and S are collinear with M between L and S. The exterior angle at vertex M, adjacent to

1. **Problem Statement:** We are given a rectangle ABCD with angles \(\angle a\) and \(\angle b\) at vertex D. The angle between the horizontal segment DC and diagonal DA is given

1. **State the problem:** We are given a triangle with sides $a=14$, $c=8$, and angle $B=64^\circ$. We need to find side $b$ using the Law of Cosines. 2. **Formula:** The Law of Co

1. **Problem Statement:** We have three scale drawings of a prism: front view (6 units wide by 6 units tall), side view (8 units wide by 3 units tall), and plan view (8 units wide

1. **Problem Statement:** We are given a circle with center $O$ and radius 2. Points $A, B, C, D, E$ lie on or inside the circle. Line segments $AC$ and $BD$ intersect at $E$. Give

1. **Problem Statement:** We have two problems involving triangles and the Pythagoras theorem.

1. **Problem statement:** We have a rectangular oil tank with dimensions 5 m (length), 3 m (height), and 2 m (width). The oil fills the tank to a depth of 1.2 m. We need to find th