Question

solve for x.
x =
(simplify your answer. type an integer or an improper fraction. use a comma to separate answers as

Explanation:

Step1: Identify similar triangles

Since $\angle QRT=\angle QPT$ and $\angle RQT=\angle PQT$ (by the markings), $\triangle QRT\sim\triangle QPT$ by the AA (angle - angle) similarity criterion.

Step2: Set up proportion

For similar triangles, the ratios of corresponding sides are equal. We have $\frac{QR}{QP}=\frac{QT}{QT + TP}$. Given $QR = 5$, $RS=6$, $TP = 6$. Let $QP=x$. Then $\frac{5}{x}=\frac{QT}{QT + 6}$. Also, using the property of similar - triangles, we can set up the proportion $\frac{QR}{QP}=\frac{RT}{PT}$. Since $\triangle QRT\sim\triangle QPT$, we know that $\frac{5}{x}=\frac{RT}{6}$. Another way is to use the fact that if we consider the ratio of the sides of the similar triangles, we have $\frac{QR}{QP}=\frac{RT}{PT}$. Let's use the property of the angle - bisector theorem. If $QT$ is the angle - bisector of $\angle PQS$, then $\frac{QR}{QP}=\frac{RT}{PT}$. We know that $\frac{5}{x}=\frac{RT}{6}$. Also, from the similarity of $\triangle QRT$ and $\triangle QPT$, we can set up the proportion $\frac{QR}{QP}=\frac{RT}{PT}=\frac{QT}{QT + TP}$. Since $\triangle QRT\sim\triangle QPT$, we have $\frac{5}{x}=\frac{RT}{6}$. Cross - multiplying gives us $5\times6=RT\times x$. But we can also use the fact that if we consider the ratio of the sides of the similar triangles formed by the angle - bisector. Let's assume the similarity of $\triangle QRS$ and $\triangle QPT$ (by AA similarity as the angles are equal). We have $\frac{QR}{QP}=\frac{RS}{PT}$. Substituting the values $QR = 5$, $RS = 6$, $PT=6$, we get $\frac{5}{x}=\frac{6}{6}$.
Cross - multiplying: $6x=5\times6$.

Step3: Solve for x

Dividing both sides of the equation $6x = 30$ by 6, we get $x = 5$.

Answer:

$5$