Question

11.
(redraw the triangles)
are the two triangles similar (using lower case letters, type yes or no)?
using capital letters, if the triangles are similar state why, otherwise type
ot
what is the length of qt?
if pt is 15 cm, what is the length of rt?

Explanation:

Sub - question 1: Are the two triangles similar?

To determine if \(\triangle PQT\) and \(\triangle RST\) are similar, we check the angles. \(\angle Q=\angle S = 90^{\circ}\), and \(\angle T\) is common to both triangles. By the AA (Angle - Angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. So the two triangles are similar.

In \(\triangle PQT\) and \(\triangle RST\), \(\angle Q=\angle S = 90^{\circ}\) (right angles) and \(\angle T\) is common to both triangles. By the AA (Angle - Angle) similarity postulate, when two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. So we state the reason as AA SIMILARITY (since \(\angle Q=\angle S\) and \(\angle T=\angle T\)).

Step 1: Let \(QS=x\) and \(QT=x + 4\). Since \(\triangle PQT\sim\triangle RST\) (by AA similarity), the ratios of corresponding sides are equal. So \(\frac{PQ}{RS}=\frac{QT}{ST}\).

We know that \(PQ = 9\space cm\), \(RS=3\space cm\), \(ST = 4\space cm\) and \(QT=QS + ST=x + 4\). Substituting into the proportion: \(\frac{9}{3}=\frac{x + 4}{4}\)

Step 2: Solve the proportion \(\frac{9}{3}=\frac{x + 4}{4}\)

Cross - multiply: \(9\times4=3\times(x + 4)\)
\(36=3x+12\)
Subtract 12 from both sides: \(3x=36 - 12=24\)
Divide both sides by 3: \(x = 8\)

Step 3: Find \(QT\)

Since \(QT=x + 4\) and \(x = 8\), then \(QT=8 + 4=12\space cm\)

Answer:

yes

Sub - question 2: State why the triangles are similar (using capital letters)