Geometry Quiz: Deriving The Triangle Area Formula

Triangle $\triangle WXY$ is shown in the plane (not necessarily right). Side $\overline{WX}$ is labeled $a$ and side $\overline{WY}$ is labeled $b$. The included angle at $W$ is labeled $C$ (so $C=\angle XWY$). A dashed altitude from $Y$ meets $\overline{WX}$ at $Z$, with $\overline{YZ}\perp\overline{WX}$. Which expression represents the area of $\triangle WXY$?

Triangle $\triangle PQR$ is shown in the plane (not necessarily right). The side $\overline{PQ}$ is labeled $a$ and the side $\overline{PR}$ is labeled $b$. The included angle at $P$ is labeled $C$ (so $C=\angle QPR$). A dashed altitude from $R$ meets $\overline{PQ}$ at $S$, with $\overline{RS}\perp\overline{PQ}$. Which expression uses the included angle correctly to give the area of $\triangle PQR$?

In triangle $UVW$, sides $UV$ and $UW$ are labeled $a$ and $b$, and the included angle $\angle VUW$ is labeled $C$. A dashed altitude from $W$ is drawn to side $UV$. Which expression represents the area of triangle $UVW$?

A non-right triangle $\triangle DEF$ is shown. Sides $DE=a$ and $DF=b$ form the included angle at $D$, labeled $C$. A dashed altitude from $E$ meets side $DF$ at a right angle. Which expression represents the area of the triangle?

A right triangle has legs of length $a$ and $b$, and the angle opposite leg $a$ measures $\theta$. If the area of this triangle is 24 square units and $\tan \theta = \frac{3}{4}$, what is the length of the hypotenuse?

Triangle $\triangle GHI$ is shown with sides $\overline{GH}=a$ and $\overline{GI}=b$, and the included angle at $G$ is labeled $C$ (so $C=\angle HGI$). A dashed altitude from $I$ meets $\overline{GH}$ at $J$ with $\overline{IJ}\perp\overline{GH}$. Which expression uses the included angle correctly to give the area of $\triangle GHI$?

In $\triangle A'B'C'$, sides $A'B'=a$ and $A'C'=b$ form the included angle at $A'$, labeled $C$. A dashed altitude from $C'$ meets side $A'B'$ at a right angle. Which expression uses the included angle correctly?

In the diagram, $\triangle JKL$ is an oblique triangle. Side $\overline{JK}$ is labeled $a$ and side $\overline{JL}$ is labeled $b$. The included angle at $J$ is labeled $C$ (so $C=\angle KJL$). A dashed altitude from $L$ meets $\overline{JK}$ at $M$, and $\overline{LM}\perp\overline{JK}$. Which conclusion follows from dropping the altitude and gives the area of $\triangle JKL$?

A surveyor measures the angle of elevation to the top of a building from two points on level ground. From point $A$, which is 100 feet from the base of the building, the angle of elevation is $53°$. From point $B$, which is 150 feet from the base on the opposite side of the building, the angle of elevation is $37°$. Assuming the ground is level and both measurements are accurate, what can be concluded about the building's height?

In the coordinate plane, right triangle $ABC$ has vertices at $A(0,0)$, $B(8,0)$, and $C(0,6)$. Point $D$ is chosen on the hypotenuse $BC$ such that when triangle $ACD$ is reflected across the $y$-axis, the reflected triangle has the same area as triangle $ABD$. What are the coordinates of point $D$?

A triangle $\triangle UVW$ is shown in the plane with sides $UV=a$ and $UW=b$ meeting at vertex $U$. The included angle at $U$ is labeled $C$, and an altitude is indicated by a dashed segment from $W$ to side $UV$. Which expression represents the area of the triangle?

In $\triangle JKL$, the sides $\overline{JK}$ and $\overline{JL}$ are labeled $a$ and $b$, and the included angle at $J$ is labeled $C$ (so $C=\angle KJL$). A dashed altitude from $L$ meets $\overline{JK}$ at $M$ and is perpendicular to $\overline{JK}$. Which conclusion follows from dropping the altitude to justify the area formula?

In the plane, triangle $\triangle STU$ has side $\overline{ST}$ labeled $a$ and side $\overline{SU}$ labeled $b$. The included angle at $S$ is labeled $C$ (so $C=\angle TSU$). A dashed altitude from $U$ meets $\overline{ST}$ at $V$, with $\overline{UV}\perp\overline{ST}$. Which relationship justifies the area formula for $\triangle STU$ using the included angle?

Triangle $\triangle RST$ is drawn in the plane and is not assumed to be right. Side $\overline{RS}$ is labeled $a$ and side $\overline{RT}$ is labeled $b$. The included angle at $R$ is labeled $C$ (so $C=\angle SRT$). A dashed altitude from $T$ meets $\overline{RS}$ at $U$, with $\overline{TU}\perp\overline{RS}$. Which claim about the area is valid?

In the diagram, $\triangle MNO$ is an oblique triangle in the plane. Side $\overline{MN}$ is labeled $a$ and side $\overline{MO}$ is labeled $b$. The included angle at $M$ is labeled $C$ (so $C=\angle NMO$). A dashed altitude from $O$ meets $\overline{MN}$ at $P$, with $\overline{OP}\perp\overline{MN}$. Which expression uses the included angle correctly?

Triangle $\triangle ABC$ is shown in the plane and is not assumed to be right. Side $\overline{AB}$ is labeled $a$ and side $\overline{AC}$ is labeled $b$. The included angle at $A$ is labeled $C$ (so $C=\angle BAC$). A dashed altitude from $C$ meets $\overline{AB}$ at $D$, with $\overline{CD}\perp\overline{AB}$. Which expression uses the included angle correctly to give the area of the triangle?

Triangle $\triangle MNP$ is shown with $\overline{MN}=a$ and $\overline{MP}=b$. The included angle at $M$ is labeled $C$ (so $C=\angle NMP$). A dashed altitude from $P$ meets $\overline{MN}$ at $Q$ with $\overline{PQ}\perp\overline{MN}$. Which expression represents the area of the triangle?

Triangle $\triangle DEF$ is shown with $\overline{DE}=a$ and $\overline{DF}=b$. The included angle at $D$ is labeled $C$ (so $C=\angle EDF$). An altitude from $F$ to line segment $\overline{DE}$ is drawn as a dashed perpendicular. Which expression represents the area of $\triangle DEF$?

Triangle $PQR$ is shown in the plane. Sides $PQ$ and $PR$ are labeled $a$ and $b$, respectively, and the included angle $\angle QPR$ is labeled $C$. A dashed altitude from $R$ meets $PQ$ at $S$ with a right-angle mark at $S$. Which expression uses the included angle correctly to give the area of triangle $PQR$?

In triangle $JKL$, side $JK$ is labeled $a$ and side $JL$ is labeled $b$. The included angle at $J$, $\angle KJL$, is labeled $C$. A dashed altitude from $L$ to $JK$ is shown. Which expression represents the area of the triangle?