ACT 1 Geo: Recognize isosceles triangles

$Q_{1}:$ A surveyor is measuring a piece of property shaped like an isosceles triangle. She knows that one side of the property is five miles long. The area of the property is $40$ square miles. How far is it from the middle of the five-mile-long side of the property to the intersection of the two other sides? other sides?

$Q_{2}:$ One leg of an isosceles right triangle is $5 m$ long. What is the length of this triangle’s hypotenuse?

$Q_{3}:$ Triangle $ABC$ in the following figure is an isosceles triangle with height $BD$. What is the value of $a$?

$Q_{4}:$ The sides of a triangle have lengths of $3, 3$, and $5.2$ meters. If one of the smaller angles in the triangle has a measure of $30$, what is the measure of the largest angle?

$Q_{5}:$ In the following figure, the points $A, B, C$, and $D$ lie along the same line. What is the difference of $x$ and $y $?

$Q_{6}:$ If $x \neq y$, which of the following expressions represents the perimeter of the triangle in the following figure in terms of $x$ and $y $?

$Q_{7}:$ The length of the shortest side of a triangle is $10$ meters, and the length of one of the longer sides is $20$ meters. If the two larger angles in the triangle are equal, what is the perimeter of the triangle in meters?

$Q_{8}:$ In Isosceles triangle, one of the following statements is false?

$Q_{9}:$ The given figure shows two isosceles triangles within a rectangle. If $b$ is half of $c$, what is the value of $a$?

$Q_{10}:$ If $XY = YZ$ in $\Delta XYZ$. If the measure of $\angle Y$ is $50$, what is the measure of $\angle Z$?

$Q_{11}:$ Which of the following could be the degree measures of the angles of an isosceles triangle?

$Q_{12}:$ Find the length of each leg of an isosceles right triangle whose hypotenuse is $6$ cm?

$Q_{13}:$ Each of the following statements is true EXCEPT:

$Q_{14}:$ In figure below, $FGHIJK$ is a regular hexagon with a perimeter of $36$. What is the length of $\overline{KG}$?

$Q_{15}:$ In figure below, $\overline{RS} = \overline{RT} = \overline{TU}$. If $m\angle RTS = 50$, then what is the measure of $\angle TRU$?

$Q_{16}:$ An equilateral triangle with sides of length $12$ is inscribed in the circle shown in figure below. What is the area of the shaded region?

$Q_{17}:$ What is the area of the quadrilateral in figure below?

$Q_{18}:$ What is the volume of the right triangular prism in figure below?

$Q_{19}:$ In the figure below, $O$ is the center of the circle. Find $m\angle O$?

$Q_{20}:$ Any triangle formed by connecting the endpoints of two radii is ...........?

$Q_{21}:$ In the figure below, if $O$ is the center of the circle, what is the value of $w$?

$Q_{22}:$ In the figure below, $\overline{PA}$and $\overline{PB}$ are tangents to circle $O$ and the measure of arc $\widehat{AXB}$ is $250$. What is $m\angle PAB$?

$Q_{23}:$ An isosceles right triangle whose legs are $6$ is rotated about one of its legs to generate a right circular cone. What is the volume of that cone?

$Q_{24}:$ If $ABC$ is isosceles and $m\angle C = 90$, which of the following must be true?

$Q_{25}:$ If in the figure below, $AB = BC = CD = AC$, what is the measure of $\angle CDE$

$Q_{26}:$ What is the area of a triangle whose sides are $5, 5,$ and $6$?

$Q_{27}:$ If a base angle of an isosceles triangle is represented by $x$, represent the number of degrees in the vertex angle.

$Q_{28}:$ In isosceles triangle $RST$, $RS = ST$. If $A$ is the midpoint of $RS$ and $B$ is the midpoint of $ST$, then