Emma wants to measure the distance across a river between two large rocks, point A and point B. From B, facing A she turns 90° and walks 79 feet to point C where there is a post. She then continues straight 23 feet to a tree stump at point D. Then, turning 90° again she walks 57 feet to point E so that she is on a straight line between the post C and the rock A on the other side of the river as shown in the diagram.
1. Triangle ABC and EDC are similar triangles. What triangle similarity theorem or postulate will prove this? Why? State which sides and angles you would use to make the argument.
2. Since the triangles are similar the corresponding sides are proportional. Write a proportion you could use to find the unknown distance from A to B using the three known distances.
3. Use “cross multiply and divide” to solve this proportion for the unknown, and state the distance from A to B to the nearest whole foot.
+9481 1.
∠ACB ≅ ∠ECD because they are vertical angles.
∠ABC ≅ ∠EDC because they are both right angles.
So by the AA similarity (postulate or theorem? I don't know..) △ABC ~ △EDC .
2.
AB / BC = ED / DC
AB / 79 = 57 / 23
3.
AB / 79 = 57 / 23
Multiply both sides of the equation by 79 .
79 * AB / 79 = 79 * 57 / 23
Simplify.
AB = 79 * 57 / 23
AB = 79 * 57 / 23
AB = 4503 / 23
AB ≈ 196 feet
+9481 1.
∠ACB ≅ ∠ECD because they are vertical angles.
∠ABC ≅ ∠EDC because they are both right angles.
So by the AA similarity (postulate or theorem? I don't know..) △ABC ~ △EDC .
2.
AB / BC = ED / DC
AB / 79 = 57 / 23
3.
AB / 79 = 57 / 23
Multiply both sides of the equation by 79 .
79 * AB / 79 = 79 * 57 / 23
Simplify.
AB = 79 * 57 / 23
AB = 79 * 57 / 23
AB = 4503 / 23
AB ≈ 196 feet
+2446 I know that the difference between postulates and theorems can perplex some of the brightest minds. I have noticed that these terms are often used interchangeably, so the confusion is understandable. The following tidbit may help you make an informed decision when dealing with the said crisis.
A theorem is a generalized mathematical statement provable via other established truths.
Some examples include the Triangle Sum Theorem, Pythagoras' Theorem, and Vertical Angles Theorem
A postulate is a generalized mathematical statement assumed as truth.
Some examples include Segment Addition Postulate (If C is between A and B, then \(AC+CB=AB\) ) and the Ruler Postulate (The points on a line can be placed in a one-to-one correspondence with the real numbers)
In this case, Angle-Angle Similarity Theorem is the appropriate name because the statement can be proved.
I guess I can attempt, to my best ability, to explain other confusing terms.
A corollary is a generalized mathematical statement heavily derived from other established theorems.
Some examples include the Corollary of the Triangle Sum Theorem (The acute angles in a right triangle are complementary) and the Corollary of the Isosceles Triangle Theorem (If a triangle is equilateral, then the triangle is equiangular).
A proposition is a mathematical result, often marked as less important
Some examples include that pi is irrational and 2 is the only even prime number.
A conjecture is an unproved mathematical statement but is believed to be true
An individual with a basic understanding of math can understand the Collatz Conjecture.
