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forsale [732]
3 years ago
10

P is the centroid of △JLN, LO⊥JN, ∠NJK≅∠JNM, OP=8, OJ=15. Find the perimeter of △JLN.

Mathematics
1 answer:
umka2103 [35]3 years ago
8 0
The complete question in the attached figure

we know that
The Centroid of a Triangle is the point where the three medians of the triangle intersect.
<span>Each median divides the triangle into two smaller triangles of equal area.
<span>and
the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex

therefore

triangles LOJ and LON are similar
ON=OJ=15 units
then
NJ=15+15-------> NJ=30 units

Let 
x--------> distance OL
so
OP=x/3
OP=8
x/3=8--------> x=8*3---------> x=24 units
OL=24 units

in the right triangle JLO
OJ=15 units
OL=24 units
JL=?
applying the Pythagorean Theorem
JL</span></span>²=OJ²+OL²--------> JL²=15²+24²------> JL=28.30 units
<span>
perimeter of triangle JLN
P=JL*2+NJ-------> P=2*28.30+30------> P=86.60 units------> P=86.6  units

the answer is
</span><span>the perimeter of △JLN is </span><span>86.6 units


</span>

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<em>Alternate solution</em>

Draw a diagonal line between the lower right inside corner and the lower right outside corner. This divides the figure into two trapezoids.

The trapezoid at lower left has bases 7 and 4 cm, and height 6-4 = 2 cm. Its area is ...

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The trapezoid at upper right has bases 6 cm and 4 cm and height 3 cm. Its area is ...

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Then the area of the figure is the sum of the areas of these trapezoids, so is ...

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_____

<em>Comment on other alternate solutions</em>

There are many other ways you can find the area of this figure. It can be divided into rectangles, triangles, or other figures of your choice. The appropriate area formulas should be used, and the resulting partial areas added or subtracted as required.

You can also let a geometry program find the area for you. (It is 26 cm².)

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