Question

in the figure above, the two lines appearing parallel are parallel. find the length of \\(\overline{eq}\\): answer

Explanation:

Step1: Identify Similar Triangles

Since \( GF \parallel RQ \), triangles \( GPF \) and \( RPQ \) are similar by the Basic Proportionality Theorem (Thales' theorem) or AA (Angle - Angle) similarity criterion (vertical angles at \( P \) and alternate interior angles due to parallel lines). For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{GP}{RP}=\frac{FP}{QP}\). Let \( EQ = x \), we know \( GP = 78 \), \( RP=91 \), \( FP = 54 \), and \( QP=x \). Wait, actually, the sides are \( GP = 78 \), \( RP = 91 \), \( FP = 54 \), and we need to find \( QP\) (which is \( EQ\)? Wait, maybe the labels: Let's assume the triangles are \( \triangle GPF \sim \triangle RPQ \), so \(\frac{GP}{RP}=\frac{FP}{QP}\).

Step2: Set up the Proportion

Substitute the known values into the proportion: \(\frac{78}{91}=\frac{54}{QP}\)

Step3: Cross - Multiply to Solve for \( QP \)

Cross - multiplying gives us \( 78\times QP=91\times54 \)

First, simplify \( \frac{78}{91}=\frac{6}{7} \) (dividing numerator and denominator by 13: \( 78\div13 = 6 \), \( 91\div13 = 7 \))

So the equation becomes \( \frac{6}{7}=\frac{54}{QP} \)

Cross - multiplying: \( 6\times QP=7\times54 \)

\( 6\times QP = 378 \)

Step4: Solve for \( QP \)

Divide both sides by 6: \( QP=\frac{378}{6}=63 \)

Answer:

63