Question

1. to find the distance across a pond from point b to point c, a surveyor drew the diagram below. the measurements he made are indicated on his diagram. use the surveyors information to determine and state the distance from point b to point c, to the nearest yard.

Explanation:

Step1: Prove similarity of triangles

Since \(\angle ADE=\angle ABC = 90^{\circ}\) and \(\angle A=\angle A\) (common angle), by the AA (Angle - Angle) similarity criterion, \(\triangle ADE\sim\triangle ABC\).

Step2: Set up proportion

For similar triangles \(\triangle ADE\) and \(\triangle ABC\), the ratios of corresponding sides are equal. That is \(\frac{AD}{AB}=\frac{AE}{AC}\). First, find \(AD\) using the Pythagorean theorem in \(\triangle ADE\): \(AD=\sqrt{AE^{2}-DE^{2}}=\sqrt{230^{2}-120^{2}}=\sqrt{(230 + 120)(230-120)}=\sqrt{350\times110}=\sqrt{38500}\approx196.21\) yd. Let \(BC = x\). Then \(\frac{AD}{AB}=\frac{DE}{BC}\). We know \(AE = 230\) yd, \(EC=85\) yd, so \(AC=AE - EC=230 - 85=145\) yd. From \(\triangle ADE\sim\triangle ABC\), we have \(\frac{196.21}{145}=\frac{120}{x}\).

Step3: Solve for \(x\)

Cross - multiply: \(196.21x=120\times145\). Then \(x=\frac{120\times145}{196.21}\approx88\) yd.

Another way:

Step1: Use the property of similar right - angled triangles

Let \(BC = x\). Since \(\triangle ADE\sim\triangle ABC\) (by AA similarity as \(\angle A\) is common and \(\angle ADE=\angle ABC = 90^{\circ}\)). The ratio of the sides of similar triangles gives \(\frac{AE}{AC}=\frac{DE}{BC}\). We know \(AE = 230\) yd, \(AC=230 - 85=145\) yd, \(DE = 120\) yd.

Step2: Substitute values into the proportion

Substitute into \(\frac{230}{145}=\frac{120}{x}\) (using the ratio of the hypotenuse and the non - hypotenuse side of similar right - angled triangles). Cross - multiply: \(230x=120\times145\).

Step3: Calculate \(x\)

\(x=\frac{120\times145}{230}=\frac{17400}{230}\approx76\) yd. Wait, error in previous approach.
Correct approach:

Step1: Use similar triangles ratio

Since \(\triangle ADE\sim\triangle ABC\) (AA similarity: \(\angle A=\angle A\), \(\angle ADE=\angle ABC = 90^{\circ}\)). The ratio of sides \( \frac{AE}{AC}=\frac{DE}{BC}\). \(AE = 230\) yd, \(AC=230 - 85 = 145\) yd, \(DE = 120\) yd.

Step2: Cross - multiply

\(230\times BC=120\times145\).

Step3: Solve for \(BC\)

\(BC=\frac{120\times145}{230}=\frac{17400}{230}=76\) yd.

Answer:

\(76\) yd