+0

# geometry

0
104
1

ACEG is a rectangle. If segment BE is 30, segment CG is 40, segment DF is 15 and ∠FDE=90∘. Find CE.

Jul 10, 2020

#1
+25600
+2

ACEG is a rectangle. If segment BE is 30, segment CG is 40, segment DF is 15 and $$\angle FDE=90^\circ$$.
Find CE.

$$\text{Let CE=x} \\ \text{Let GE=y} \\ \text{Let AB=z} \\ \text{Let BC=y-z}$$

$$\begin{array}{|lrcll|} \hline 1): & x^2 + y^2 &=& 40^2 \\ & x &=& \sqrt{40^2-y^2} \\ \hline 2): & x^2+(y-z)^2 &=& 30^2 \quad | \quad x^2 = 40^2-y^2 \\ & 40^2-y^2+(y-z)^2 &=& 30^2 \\ & y^2 - (y-z)^2 &=& 40^2-30^2 \\ & \mathbf{y^2 - (y-z)^2} &=& \mathbf{700} \qquad (1) \\ \hline \end{array}$$

$$\begin{array}{|lrcll|} \hline 3): & \text{area \triangle  CDE} &=& \text{area \triangle  CFE} + \text{area \triangle  FEG} \\ & \dfrac{xy}{2} &=& \dfrac{x*15}{2} + \dfrac{y*DE}{2} \\ & xy &=& x*15 + y*DE \\ & && \boxed{\dfrac{DE}{15} =\dfrac{x}{y-z}\\ DE = \dfrac{15x}{y-z} } \\ & xy &=& x*15 + y*\left( \dfrac{15x}{y-z} \right) \quad | \quad : x \\ & y &=& 15 + \left( \dfrac{15y}{y-z} \right) \\ & (y-15)(y-z) &=& 15y \\ & y-z &=& \dfrac{15y}{y-15} \\ & \mathbf{(y-z)^2} &=& \mathbf{\dfrac{225y^2}{(y-15)^2}} \qquad (2) \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{y^2 - (y-z)^2} &=& \mathbf{700} \quad | \quad \mathbf{(y-z)^2=\dfrac{225y^2}{(y-15)^2}} \\ y^2 - \dfrac{225y^2}{(y-15)^2} &=& 700 \quad | \quad *(y-15)^2 \\\\ y^2(y-15)^2 - 225y^2 &=& 700(y-15)^2 \\ y^2(y^2-30y+225) - 225y^2 &=& 700(y^2-30y+225) \\ y^4-30y^3+225y^2-225y^2 &=& 700y^2-700*30y+700*225 \\ y^4-30y^3 &=& 700y^2-700*30y+700*225 \\ \mathbf{ y^4-30y^3 -700y^2+700*30y-700*225} &=& \mathbf{0} \\ \Rightarrow \mathbf{y}&=&\mathbf{36.6659934977741} \qquad \text{WolframAlpha} \\ \hline x &=& \sqrt{40^2-y^2} \\ x &=& \sqrt{40^2-36.6659934977741^2} \\ \mathbf{x}&=&\mathbf{15.9876490087} \\ \hline \end{array}$$

CE is $$\approx \mathbf{16}$$

Jul 10, 2020
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7091859579086304, "perplexity": 10608.998146765343}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141747774.97/warc/CC-MAIN-20201205104937-20201205134937-00059.warc.gz"}