2023年中考數(shù)學(xué)真題分項(xiàng)匯編專(zhuān)題16 等腰三角形與直角三角形（原卷版）

專(zhuān)題16等腰三角形與直角三角形（共26道）一、單選題1．（2023·江蘇徐州·統(tǒng)考中考真題）如圖，在SKIPIF1<0中，SKIPIF1<0為SKIPIF1<0的中點(diǎn)．若點(diǎn)SKIPIF1<0在邊SKIPIF1<0上，且SKIPIF1<0，則SKIPIF1<0的長(zhǎng)為（

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A．1 B．2 C．1或SKIPIF1<0 D．1或22．（2023·甘肅蘭州·統(tǒng)考中考真題）如圖，在矩形SKIPIF1<0中，點(diǎn)E為SKIPIF1<0延長(zhǎng)線(xiàn)上一點(diǎn)，F(xiàn)為SKIPIF1<0的中點(diǎn)，以B為圓心，SKIPIF1<0長(zhǎng)為半徑的圓弧過(guò)SKIPIF1<0與SKIPIF1<0的交點(diǎn)G，連接SKIPIF1<0．若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）

A．2 B．2.5 C．3 D．3.53．（2023·北京·統(tǒng)考中考真題）如圖，點(diǎn)A、B、C在同一條線(xiàn)上，點(diǎn)B在點(diǎn)A，C之間，點(diǎn)D，E在直線(xiàn)AC同側(cè)，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，連接DE，設(shè)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，給出下面三個(gè)結(jié)論：①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0；

上述結(jié)論中，所有正確結(jié)論的序號(hào)是（

）A．①② B．①③ C．②③ D．①②③4．（2023·江蘇無(wú)錫·統(tǒng)考中考真題）如圖SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0為SKIPIF1<0中點(diǎn)，若點(diǎn)SKIPIF1<0為直線(xiàn)SKIPIF1<0下方一點(diǎn)，且SKIPIF1<0與SKIPIF1<0相似，則下列結(jié)論：①若SKIPIF1<0，SKIPIF1<0與SKIPIF1<0相交于SKIPIF1<0，則點(diǎn)SKIPIF1<0不一定是SKIPIF1<0的重心；②若SKIPIF1<0，則SKIPIF1<0的最大值為SKIPIF1<0；③若SKIPIF1<0，則SKIPIF1<0的長(zhǎng)為SKIPIF1<0；④若SKIPIF1<0，則當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0取得最大值．其中正確的為（

）

A．①④ B．②③ C．①②④ D．①③④5．（2023·浙江·統(tǒng)考中考真題）如圖，在四邊形SKIPIF1<0中，SKIPIF1<0，以SKIPIF1<0為腰作等腰直角三角形SKIPIF1<0，頂點(diǎn)SKIPIF1<0恰好落在SKIPIF1<0邊上，若SKIPIF1<0，則SKIPIF1<0的長(zhǎng)是（

）

A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．16．（2023·四川眉山·統(tǒng)考中考真題）如圖，在正方形SKIPIF1<0中，點(diǎn)E是SKIPIF1<0上一點(diǎn)，延長(zhǎng)SKIPIF1<0至點(diǎn)F，使SKIPIF1<0，連結(jié)SKIPIF1<0，SKIPIF1<0交SKIPIF1<0于點(diǎn)K，過(guò)點(diǎn)A作SKIPIF1<0，垂足為點(diǎn)H，交SKIPIF1<0于點(diǎn)G，連結(jié)SKIPIF1<0．下列四個(gè)結(jié)論：①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0；④SKIPIF1<0．其中正確結(jié)論的個(gè)數(shù)為（

）

A．1個(gè) B．2個(gè) C．3個(gè) D．4個(gè)二、填空題7．（2023·湖南·統(tǒng)考中考真題）七巧板是我國(guó)民間廣為流傳的一種益智玩具，某同學(xué)用邊長(zhǎng)為SKIPIF1<0的正方形紙板制作了一副七巧板（如圖），由5個(gè)等腰直角三角形，1個(gè)正方形和1個(gè)平行四邊形組成．則圖中陰影部分的面積為_(kāi)_________SKIPIF1<0．

8．（2023·天津·統(tǒng)考中考真題）如圖，在邊長(zhǎng)為3的正方形SKIPIF1<0的外側(cè)，作等腰三角形SKIPIF1<0，SKIPIF1<0．

（1）SKIPIF1<0的面積為_(kāi)_______；（2）若F為SKIPIF1<0的中點(diǎn)，連接SKIPIF1<0并延長(zhǎng)，與SKIPIF1<0相交于點(diǎn)G，則SKIPIF1<0的長(zhǎng)為_(kāi)_______．9．（2023·河南·統(tǒng)考中考真題）矩形SKIPIF1<0中，M為對(duì)角線(xiàn)SKIPIF1<0的中點(diǎn)，點(diǎn)N在邊SKIPIF1<0上，且SKIPIF1<0．當(dāng)以點(diǎn)D，M，N為頂點(diǎn)的三角形是直角三角形時(shí)，SKIPIF1<0的長(zhǎng)為_(kāi)_____．10．（2023·湖北·統(tǒng)考中考真題）如圖，SKIPIF1<0和SKIPIF1<0都是等腰直角三角形，SKIPIF1<0，點(diǎn)SKIPIF1<0在SKIPIF1<0內(nèi)，SKIPIF1<0，連接SKIPIF1<0交SKIPIF1<0于點(diǎn)SKIPIF1<0交SKIPIF1<0于點(diǎn)SKIPIF1<0，連接SKIPIF1<0．給出下面四個(gè)結(jié)論：①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0；④SKIPIF1<0．其中所有正確結(jié)論的序號(hào)是_________．11．（2023·山東·統(tǒng)考中考真題）如圖，SKIPIF1<0是邊長(zhǎng)為6的等邊三角形，點(diǎn)SKIPIF1<0在邊SKIPIF1<0上，若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0_________．

12．（2023·山東日照·統(tǒng)考中考真題）如圖，矩形SKIPIF1<0中，SKIPIF1<0，點(diǎn)P在對(duì)角線(xiàn)SKIPIF1<0上，過(guò)點(diǎn)P作SKIPIF1<0，交邊SKIPIF1<0于點(diǎn)M，N，過(guò)點(diǎn)M作SKIPIF1<0交SKIPIF1<0于點(diǎn)E，連接SKIPIF1<0．下列結(jié)論：①SKIPIF1<0；②四邊形SKIPIF1<0的面積不變；③當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0；④SKIPIF1<0的最小值是20．其中所有正確結(jié)論的序號(hào)是__________．

13．（2023·四川遂寧·統(tǒng)考中考真題）如圖，以SKIPIF1<0的邊SKIPIF1<0、SKIPIF1<0為腰分別向外作等腰直角SKIPIF1<0、SKIPIF1<0，連結(jié)SKIPIF1<0、SKIPIF1<0、SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0的直線(xiàn)SKIPIF1<0分別交線(xiàn)段SKIPIF1<0、SKIPIF1<0于點(diǎn)SKIPIF1<0、SKIPIF1<0，以下說(shuō)法：①當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0；②SKIPIF1<0；③若SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0；④當(dāng)直線(xiàn)SKIPIF1<0時(shí)，點(diǎn)SKIPIF1<0為線(xiàn)段SKIPIF1<0的中點(diǎn)．正確的有_________．(填序號(hào))

14．（2023·四川眉山·統(tǒng)考中考真題）如圖，在平面直角坐標(biāo)系SKIPIF1<0中，點(diǎn)B的坐標(biāo)為SKIPIF1<0，過(guò)點(diǎn)B分別作x軸、y軸的垂線(xiàn)，垂足分別為點(diǎn)C、點(diǎn)A，直線(xiàn)SKIPIF1<0與SKIPIF1<0交于點(diǎn)D．與y軸交于點(diǎn)E．動(dòng)點(diǎn)M在線(xiàn)段SKIPIF1<0上，動(dòng)點(diǎn)N在直線(xiàn)SKIPIF1<0上，若SKIPIF1<0是以點(diǎn)N為直角頂點(diǎn)的等腰直角三角形，則點(diǎn)M的坐標(biāo)為_(kāi)_______

15．（2023·江蘇蘇州·統(tǒng)考中考真題）如圖，SKIPIF1<0．過(guò)點(diǎn)SKIPIF1<0作SKIPIF1<0，延長(zhǎng)SKIPIF1<0到SKIPIF1<0，使SKIPIF1<0，連接SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0________________．（結(jié)果保留根號(hào)）

16．（2023·山西·統(tǒng)考中考真題）如圖，在四邊形SKIPIF1<0中，SKIPIF1<0，對(duì)角線(xiàn)SKIPIF1<0相交于點(diǎn)SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0的長(zhǎng)為_(kāi)_________．

17．（2023·湖北十堰·統(tǒng)考中考真題）在某次數(shù)學(xué)探究活動(dòng)中，小明將一張斜邊為4的等腰直角三角形SKIPIF1<0硬紙片剪切成如圖所示的四塊（其中D，E，F(xiàn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的中點(diǎn)，G，H分別為SKIPIF1<0，SKIPIF1<0的中點(diǎn)），小明將這四塊紙片重新組合拼成四邊形（相互不重疊，不留空隙），則所能拼成的四邊形中周長(zhǎng)的最小值為_(kāi)___________，最大值為_(kāi)__________________．

三、解答題18．（2023·北京·統(tǒng)考中考真題）在SKIPIF1<0中、SKIPIF1<0，SKIPIF1<0于點(diǎn)M，D是線(xiàn)段SKIPIF1<0上的動(dòng)點(diǎn)（不與點(diǎn)M，C重合），將線(xiàn)段SKIPIF1<0繞點(diǎn)D順時(shí)針旋轉(zhuǎn)SKIPIF1<0得到線(xiàn)段SKIPIF1<0．

(1)如圖1，當(dāng)點(diǎn)E在線(xiàn)段SKIPIF1<0上時(shí)，求證：D是SKIPIF1<0的中點(diǎn)；(2)如圖2，若在線(xiàn)段SKIPIF1<0上存在點(diǎn)F（不與點(diǎn)B，M重合）滿(mǎn)足SKIPIF1<0，連接SKIPIF1<0，SKIPIF1<0，直接寫(xiě)出SKIPIF1<0的大小，并證明．19．（2023·黑龍江·統(tǒng)考中考真題）如圖①，SKIPIF1<0和SKIPIF1<0是等邊三角形，連接SKIPIF1<0，點(diǎn)F，G，H分別是SKIPIF1<0和SKIPIF1<0的中點(diǎn)，連接SKIPIF1<0．易證：SKIPIF1<0．若SKIPIF1<0和SKIPIF1<0都是等腰直角三角形，且SKIPIF1<0，如圖②：若SKIPIF1<0和SKIPIF1<0都是等腰三角形，且SKIPIF1<0，如圖③：其他條件不變，判斷SKIPIF1<0和SKIPIF1<0之間的數(shù)量關(guān)系，寫(xiě)出你的猜想，并利用圖②或圖③進(jìn)行證明．

20．（2023·黑龍江齊齊哈爾·統(tǒng)考中考真題）綜合與實(shí)踐數(shù)學(xué)模型可以用來(lái)解決一類(lèi)問(wèn)題，是數(shù)學(xué)應(yīng)用的基本途徑．通過(guò)探究圖形的變化規(guī)律，再結(jié)合其他數(shù)學(xué)知識(shí)的內(nèi)在聯(lián)系，最終可以獲得寶貴的數(shù)學(xué)經(jīng)驗(yàn)，并將其運(yùn)用到更廣闊的數(shù)學(xué)天地．

(1)發(fā)現(xiàn)問(wèn)題：如圖1，在SKIPIF1<0和SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，連接SKIPIF1<0，SKIPIF1<0，延長(zhǎng)SKIPIF1<0交SKIPIF1<0于點(diǎn)SKIPIF1<0．則SKIPIF1<0與SKIPIF1<0的數(shù)量關(guān)系：______，SKIPIF1<0______SKIPIF1<0；(2)類(lèi)比探究：如圖2，在SKIPIF1<0和SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，連接SKIPIF1<0，SKIPIF1<0，延長(zhǎng)SKIPIF1<0，SKIPIF1<0交于點(diǎn)SKIPIF1<0．請(qǐng)猜想SKIPIF1<0與SKIPIF1<0的數(shù)量關(guān)系及SKIPIF1<0的度數(shù)，并說(shuō)明理由；(3)拓展延伸：如圖3，SKIPIF1<0和SKIPIF1<0均為等腰直角三角形，SKIPIF1<0，連接SKIPIF1<0，SKIPIF1<0，且點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0在一條直線(xiàn)上，過(guò)點(diǎn)SKIPIF1<0作SKIPIF1<0，垂足為點(diǎn)SKIPIF1<0．則SKIPIF1<0，SKIPIF1<0，SKIPIF1<0之間的數(shù)量關(guān)系：______；(4)實(shí)踐應(yīng)用：正方形SKIPIF1<0中，SKIPIF1<0，若平面內(nèi)存在點(diǎn)SKIPIF1<0滿(mǎn)足SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0______．21．（2023·四川成都·統(tǒng)考中考真題）探究式學(xué)習(xí)是新課程倡導(dǎo)的重要學(xué)習(xí)方式，某興趣小組擬做以下探究．在SKIPIF1<0中，SKIPIF1<0，D是SKIPIF1<0邊上一點(diǎn)，且SKIPIF1<0（n為正整數(shù)），E是SKIPIF1<0邊上的動(dòng)點(diǎn)，過(guò)點(diǎn)D作SKIPIF1<0的垂線(xiàn)交直線(xiàn)SKIPIF1<0于點(diǎn)F．【初步感知】(1)如圖1，當(dāng)SKIPIF1<0時(shí)，興趣小組探究得出結(jié)論：SKIPIF1<0，請(qǐng)寫(xiě)出證明過(guò)程．【深入探究】(2)①如圖2，當(dāng)SKIPIF1<0，且點(diǎn)F在線(xiàn)段SKIPIF1<0上時(shí)，試探究線(xiàn)段SKIPIF1<0之間的數(shù)量關(guān)系，請(qǐng)寫(xiě)出結(jié)論并證明；②請(qǐng)通過(guò)類(lèi)比、歸納、猜想，探究出線(xiàn)段SKIPIF1<0之間數(shù)量關(guān)系的一般結(jié)論（直接寫(xiě)出結(jié)論，不必證明）【拓展運(yùn)用】(3)如圖3，連接SKIPIF1<0，設(shè)SKIPIF1<0的中點(diǎn)為M．若SKIPIF1<0，求點(diǎn)E從點(diǎn)A運(yùn)動(dòng)到點(diǎn)C的過(guò)程中，點(diǎn)M運(yùn)動(dòng)的路徑長(zhǎng)（用含n的代數(shù)式表示）．22．（2023·吉林長(zhǎng)春·統(tǒng)考中考真題）如圖①．在矩形SKIPIF1<0．SKIPIF1<0，點(diǎn)SKIPIF1<0在邊SKIPIF1<0上，且SKIPIF1<0．動(dòng)點(diǎn)SKIPIF1<0從點(diǎn)SKIPIF1<0出發(fā)，沿折線(xiàn)SKIPIF1<0以每秒SKIPIF1<0個(gè)單位長(zhǎng)度的速度運(yùn)動(dòng)，作SKIPIF1<0，SKIPIF1<0交邊SKIPIF1<0或邊SKIPIF1<0于點(diǎn)SKIPIF1<0，連續(xù)SKIPIF1<0．當(dāng)點(diǎn)SKIPIF1<0與點(diǎn)SKIPIF1<0重合時(shí)，點(diǎn)SKIPIF1<0停止運(yùn)動(dòng)．設(shè)點(diǎn)SKIPIF1<0的運(yùn)動(dòng)時(shí)間為SKIPIF1<0秒．（SKIPIF1<0）

(1)當(dāng)點(diǎn)SKIPIF1<0和點(diǎn)SKIPIF1<0重合時(shí)，線(xiàn)段SKIPIF1<0的長(zhǎng)為_(kāi)_________；(2)當(dāng)點(diǎn)SKIPIF1<0和點(diǎn)SKIPIF1<0重合時(shí)，求SKIPIF1<0；(3)當(dāng)點(diǎn)SKIPIF1<0在邊SKIPIF1<0上運(yùn)動(dòng)時(shí)，SKIPIF1<0的形狀始終是等腰直角三角形．如圖②．請(qǐng)說(shuō)明理由；(4)作點(diǎn)SKIPIF1<0關(guān)于直線(xiàn)SKIPIF1<0的對(duì)稱(chēng)點(diǎn)SKIPIF1<0，連接SKIPIF1<0、SKIPIF1<0，當(dāng)四邊形SKIPIF1<0和矩形SKIPIF1<0重疊部分圖形為軸對(duì)稱(chēng)四邊形時(shí)，直接寫(xiě)出SKIPIF1<0的取值范圍．23．（2023·甘肅武威·統(tǒng)考中考真題）【模型建立】（1）如圖1，SKIPIF1<0和SKIPIF1<0都是等邊三角形，點(diǎn)SKIPIF1<0關(guān)于SKIPIF1<0的對(duì)稱(chēng)點(diǎn)SKIPIF1<0在SKIPIF1<0邊上．①求證：SKIPIF1<0；②用等式寫(xiě)出線(xiàn)段SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的數(shù)量關(guān)系，并說(shuō)明理由．【模型應(yīng)用】（2）如圖2，SKIPIF1<0是直角三角形，SKIPIF1<0，SKIPIF1<0，垂足為SKIPIF1<0，點(diǎn)SKIPIF1<0關(guān)于SKIPIF1<0的對(duì)稱(chēng)點(diǎn)SKIPIF1<0在SKIPIF1<0邊上．用等式寫(xiě)出線(xiàn)段SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的數(shù)量關(guān)系，并說(shuō)明理由．【模型遷移】（3）在（2）的條件下，若SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0的值．

24．（2023·重慶·統(tǒng)考中考真題）如圖，在等邊SKIPIF1<0中，SKIPIF1<0于點(diǎn)SKIPIF1<0，SKIPIF1<0為線(xiàn)段SKIPIF1<0上一動(dòng)點(diǎn)（不與SKIPIF1<0，SKIPIF1<0重合），連接SKIPIF1<0，SKIPIF1<0，將SKIPIF1<0繞點(diǎn)SKIPIF1<0順時(shí)針旋轉(zhuǎn)SKIPIF1<0得到線(xiàn)段SKIPIF1<0，連接SKIPIF1<0．(1)如圖1，求證：SKIPIF1<0；(2)如圖2，連接SKIPIF1<0交SKIPIF1<0于點(diǎn)SKIPIF1<0，連接SKIPIF1<0，SKIPIF1<0，SKIPIF1<0與SKIPIF1<0所在直線(xiàn)交于點(diǎn)SKIPIF1<0，求證：SKIPIF1<0；(3)如圖3，連接SKIPIF1<0交SKIPIF1<0于點(diǎn)SKIPIF1<0，連接SKIPIF1<0，SKIPIF1<0，將SKIPIF1<0沿SKIPIF1<0所