The distance across the river shown in the figure can be...

The distance across the river shown in the figure can be found without measuring angles. Two points $B$ and $C$ on the opposite shore are selected, and line segments $A B$ and $A C$ are extended as shown. Points $D$ and $E$ are chosen as indicated, and distances $B C, B D, B E, C D$ and $C E$ are then measured. Suppose that $B C=184 \mathrm{ft}$, $B D=102 \mathrm{ft}, B E=218 \mathrm{ft}, C D=236 \mathrm{ft},$ and $C E=80 \mathrm{ft}$ . (a) Approximate the distances $A B$ and $A C$. (b) Approximate the shortest distance across the river from point $A$ (FIGURE CAN'T COPY)

The distance across the river shown in the figure can be found without measuring angles. Two points $B$ and $C$ on the opposite shore are selected, and line segments $A B$ and $A C$ are extended as shown. Points $D$ and $E$ are chosen as indicated, and distances $B C, B D, B E, C D$ and $C E$ are then measured. Suppose that $B C=184 \mathrm{ft}$, $B D=102 \mathrm{ft}, B E=218 \mathrm{ft}, C D=236 \mathrm{ft},$ and $C E=80 \mathrm{ft}$ .
(a) Approximate the distances $A B$ and $A C$.
(b) Approximate the shortest distance across the river from point $A$
(FIGURE CAN'T COPY)

Key Concepts

Similar Triangles

Similar triangles are triangles that share the same shape, meaning their corresponding angles are equal and their corresponding sides are in proportion. This principle allows one to create relationships between known measurements and unknown distances by comparing ratios of corresponding sides in different triangles formed through the construction.

Proportional Reasoning

Proportional reasoning involves setting up and solving ratios derived from the corresponding sides of similar triangles. It is a critical step in using measured segments to indirectly determine unknown lengths, as the constant ratio between sides provides a way to scale measured distances to the desired measurement.

Indirect Measurement

Indirect measurement is a technique used to determine the dimensions of an object or distance that is difficult to measure directly. In problems where direct measurement is impractical, techniques like constructing similar triangles and applying proportional reasoning enable the calculation of distances using other accessible measurements.

Transcript

00:01 This problem is kind of a difficult one, and i hope i did it right.

00:06 I think so.

00:07 The answer makes sense.

00:09 So we want to figure out this.

00:12 What are they actually asked? the distance is here.

00:16 So the distance across the river is shown in the figure can be found without measuring angles.

00:21 So we need two points, b and c on opposite shore.

00:25 Okay, so here and here, are selected.

00:32 Points and line a, b, and a, c.

00:36 Oh, so, wait a minute, b and c.

00:38 So these total points here are given here.

00:41 And then we extend those to out here, and the line segments are extended as shown.

00:48 So we have point a here, then we extend them here, and we extend it out here.

00:52 And then we have these crosses here that we measure.

00:56 So we can measure this, you can measure this, we can measure this.

01:00 We can measure this, this, and we can also measure that, but they didn't give us that information.

01:07 And with that information, we can figure out what this length is and what this length is.

01:13 Now, basically what you'd have to do, that you have to make sure you cite to something here.

01:20 So you can cite along and keep this, cite something common over here, like if there's a tree or something over here.

01:31 So they tell us bc is 185 184 feet and bd is 102 feet b .e is 218 feet.

01:42 C .d.

01:43 Is 236 feet and c .e is 80 feet.

01:50 So what we need to do, what i decided to do is i was going to figure out if we can figure out what, let's see here.

02:00 We can look at this triangle here and figure out this angle and then this angle here is 180 minus that angle.

02:11 And then we can look at, let's see here, this triangle here, and find this angle, and then this angle is 180 minus that.

02:21 So we have, i call theta 1 the angle d, b, c...

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