We are asked to find the distance across the stream using the fact that Mr. Turner uses a carpenter's square to do this.
Let's notice that the sum of the measures of ∠ MOK and ∠ KOP is equal to 90^(∘), as together they form a right angle.
m∠ MOK+m∠ KOP=90^(∘)
⇓
m∠ MOK = 90^(∘)-m∠ KOPNext, recall that the sum of all angle measures in each triangle is equal to 180^(∘). Using this, we can write an equation for △ OMK. Then we will substitute the values we know.
Since △ OMK and △ POK are right triangles and they have one more congruent angle, they are similar by Angle-Angle Similarity Theorem. Let's recall that in similar triangles corresponding sides are proportional. With this information, we can write the ratio.
MK/OK=OK/KP
We are given that MK is 1.5 ft and OK is 4.5 ft. Therefore, to solve for KP we will substitute the given lengths into the above proportion.
The length of MO is approximately 4.74. Let's add this information to our picture.
Next, let's notice that △ POM and △ OKM are similar by the Angle-Angle Similarity Theorem, as they are both right triangles and they share ∠ M. Using this fact, we can create a proportion.
MP/MO=MO/MK
Let's substitute appropriate values and solve for MP.
Finally, we can evaluate the length of KP by subtracting the length of MK from the length of MP.
KP=MP-MK
KP=15- 1.5=13.5
The distance across the stream is 13.5 ft.
