+0

# Geometry Help

0
4
1
+8

Find DB in the diagram below.

[asy] size(250); pair A,B,C,D; A=origin; B=(4,0); C=3dir(30); D=(2.3,0); draw(C--A--B--C--D); dot("\$A\$",A,SW); dot("\$B\$",B,SE); dot("\$C\$",C,N); dot("\$D\$",D,S); label("\$2\$",A--D,S); label("\$30^\circ\$",A,6dir(15)); label("\$45^\circ\$",B,4dir(158)); label("\$60^\circ\$",C,5dir(-75)); [/asy]

Jun 13, 2024

#1
+814
0

Identify Right Triangles: Since ∠ABC=45∘, triangle ABC is a 45-45-90 triangle. This means ∠BAC=45∘ and ∠ACB=90∘. Additionally, since ∠CAD=30∘ and ∠ACB=90∘, triangle ACD is a 30-60-90 triangle.

Find AC in Triangle ACD: Because triangle ACD is a 30-60-90 triangle, we know that the ratio of side lengths is constant (i.e., AC/AD = 3​). We are given that AD = 2, so:

Find AB in Triangle ABC: Since triangle ABC is a 45-45-90 triangle, we know that the ratio of side lengths is also constant (i.e., AB/AC = 1). We found AC in the previous step, so:

AB = AC = 2√3

Find BD: Now that we know AB, we can find BD by subtracting AD from AB:

BD = AB - AD = 2√3 - 2

Therefore, the length of BD is 2*sqrt(3) - 2.

Jun 13, 2024