Answer: 37
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Step-by-step explanation
The diagram provides us with three facts:
- AB = BC because of the tick marks.
- Angle DBC = angle ABD because both are right angles (aka 90 degree angles).
- BD = BD because of the reflexive property.
Because of those facts, we can prove triangle BDA is congruent to triangle BDC by use of the SAS theorem.
SAS = side angle side
Notice: facts 1 and 3 are the two "S" letters of "SAS", while fact 2 handles the "A" of "SAS". Another thing to notice is that the angles mentioned are between the congruent sides. The order of the facts I've mentioned hopefully help show how they align with SAS.
The key takeaway from these congruent triangles is that AD = CD because of CPCTC which stands for "Corresponding Parts of Congruent Triangles are Congruent". It's like saying "two houses are the same, so their front doors must be identical".
In short, the triangles are mirror clones of each other. The line of reflection is segment BD.
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The previous section concludes with segments AD and CD being the same length. Let's use this to find x.
AD = CD
4x+5 = 3x+13
4x-3x = 13-5
x = 8
Then we can find each segment length.
- CD = 3x+13 = 3*8+13 = 24+13 = 37
- AD = 4x+5 = 4*8+5 = 32+5 = 37
Both CD and AD are 37 units long, which helps confirm the answer is correct.